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Stars and Bars - Intermediate Level: tricky scenarios handling Stars and Bars INTERMEDIATE

This expert challenge 📈 worksheet focuses on Stars and Bars - a key topic in Permutation Combination. You'll solve 20 intermediate-level problems (Worksheet 5 of 10). The primary focus is on tricky scenarios handling. Master how to solve stars and bars, stars and bars tricks, and stars and bars shortcut methods through systematic practice.

📝 Worksheet 5 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Intermediate level

What you'll learn in this worksheet:
Your progress through Stars and Bars
Worksheet 5 of 10 (44% complete)

Question 1

In how many ways can 15 identical cookies be given to 4 distinct children (where each recipient can receive zero or more)? Or, find the number of 4 distinct children.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 2

In how many ways can 13 identical cookies be given to 3 distinct children (where each recipient can receive zero or more)? Or, find the number of 3 distinct children.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 3

In how many ways can 14 similar coins be given to 5 different banks (where each recipient can receive zero or more)? Or, find the number of 5 different banks.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 4

In how many ways can 12 similar coins be given to 4 different banks (where each recipient can receive zero or more)? Or, find the number of 4 different banks.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 5

In how many ways can 11 identical cookies be given to 5 distinct children (where each recipient can receive zero or more)? Or, find the number of 5 distinct children.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 6

In how many ways can 10 similar coins be given to 5 different banks (where each recipient can receive zero or more)? Or, find the number of 5 different banks.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 7

In how many ways can 12 identical cookies be given to 3 distinct children (where each recipient can receive zero or more)? Or, find the number of 3 distinct children.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 8

In how many ways can 10 similar coins be given to 4 different banks (where each recipient can receive zero or more)? Or, find the number of 4 different banks.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 9

In how many ways can non-negative integer solutions be given to the equation x1 + x2 + ... + x5 = 15 (where each recipient can receive zero or more)? Or, find the number of the equation x1 + x2 + ... + x5 = 15.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 10

In how many ways can 13 identical cookies be given to 5 distinct children (where each recipient can receive zero or more)? Or, find the number of 5 distinct children.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 11

In how many ways can 15 similar coins be given to 4 different banks (where each recipient can receive zero or more)? Or, find the number of 4 different banks.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 12

In how many ways can 13 similar coins be given to 3 different banks (where each recipient can receive zero or more)? Or, find the number of 3 different banks.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 13

In how many ways can 13 identical cookies be given to 4 distinct children (where each recipient can receive zero or more)? Or, find the number of 4 distinct children.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 14

In how many ways can non-negative integer solutions be given to the equation x1 + x2 + ... + x4 = 10 (where each recipient can receive zero or more)? Or, find the number of the equation x1 + x2 + ... + x4 = 10.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 15

In how many ways can 15 identical cookies be given to 5 distinct children (where each recipient can receive zero or more)? Or, find the number of 5 distinct children.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 16

In how many ways can non-negative integer solutions be given to the equation x1 + x2 + ... + x4 = 13 (where each recipient can receive zero or more)? Or, find the number of the equation x1 + x2 + ... + x4 = 13.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 17

In how many ways can 10 identical cookies be given to 4 distinct children (where each recipient can receive zero or more)? Or, find the number of 4 distinct children.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 18

In how many ways can non-negative integer solutions be given to the equation x1 + x2 + ... + x4 = 13 (where each recipient can receive zero or more)? Or, find the number of the equation x1 + x2 + ... + x4 = 13.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 19

In how many ways can 15 similar coins be given to 4 different banks (where each recipient can receive zero or more)? Or, find the number of 4 different banks.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).

Question 20

In how many ways can 11 similar coins be given to 3 different banks (where each recipient can receive zero or more)? Or, find the number of 3 different banks.
Step-by-Step Solution (Stars and Bars):

Concept: This problem is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 + \dots + x_{k} = {n}$. This is a distribution problem of identical items ({n} 'stars') into distinct containers ({k} 'bins') using {k-1} separators ('bars').

Formula: The number of solutions is $\text{C}(n + k - 1, k - 1)$.
- $n$ = number of identical items (stars) = {n}
- $k$ = number of distinct recipients (bins) = {k}

Calculation:
Total arrangements = $\text{C}({n} + {k} - 1, {k} - 1)$
= $\text{C}({n + k - 1}, {k - 1})$
= {answer}

Key Distinction:
- Identical Items, Distinct Boxes (Stars and Bars): $\text{C}(n+k-1, k-1)$
- Distinct Items, Distinct Boxes (Distribution): $k^n$

Verification: This method guarantees that all solutions are non-negative ($x_i \ge 0$).
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