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Permutations with Identical Objects - Expert Level: conceptual clarity Permutations with Identical Objects EXPERT

This skill evaluation ⚡ worksheet focuses on Permutations with Identical Objects - a key topic in Permutation Combination. You'll solve 20 expert-level problems (Worksheet 9 of 10). The primary focus is on conceptual clarity. Master permutations with identical objects ssc cgl, permutations with identical objects reasoning tricks, and fast permutations with identical objects solving through systematic practice.

📝 Worksheet 9 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Expert level

What you'll learn in this worksheet:
Your progress through Permutations with Identical Objects
Worksheet 9 of 10 (88% complete)

Question 1

A word has 7 letters: 2 as, 2 bs, 3 cs. How many distinct ways can these letters be arranged?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 7
- Distribution: Type 1: 2, Type 2: 2, Type 3: 3

Step 1 - Total arrangements if all were distinct:
7! = 5040

Step 2 - Account for identical objects:
7! = 5040 / 2! = 2 / 2! = 2 / 3! = 6

Final Calculation:
= 210

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 7! = 5040.

Question 2

You have 8 books: 2 copies of book type A, 3 copies of book type B, 3 copies of book type C. How many distinct ways can these books be lined on a shelf?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 8
- Distribution: Type 1: 2, Type 2: 3, Type 3: 3

Step 1 - Total arrangements if all were distinct:
8! = 40320

Step 2 - Account for identical objects:
8! = 40320 / 2! = 2 / 3! = 6 / 3! = 6

Final Calculation:
= 560

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 8! = 40320.

Question 3

You have 14 books: 4 copies of book type A, 2 copies of book type B, 4 copies of book type C, 3 copies of book type D, 1 copies of book type E. How many distinct ways can these books be lined on a shelf?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 14
- Distribution: Type 1: 4, Type 2: 2, Type 3: 4, Type 4: 3, Type 5: 1

Step 1 - Total arrangements if all were distinct:
14! = 87178291200

Step 2 - Account for identical objects:
14! = 87178291200 / 4! = 24 / 2! = 2 / 4! = 24 / 3! = 6

Final Calculation:
= 12612600

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 14! = 87178291200.

Question 4

You have 7 books: 2 copies of book type A, 2 copies of book type B, 2 copies of book type C, 1 copies of book type D. How many distinct ways can these books be lined on a shelf?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 7
- Distribution: Type 1: 2, Type 2: 2, Type 3: 2, Type 4: 1

Step 1 - Total arrangements if all were distinct:
7! = 5040

Step 2 - Account for identical objects:
7! = 5040 / 2! = 2 / 2! = 2 / 2! = 2

Final Calculation:
= 630

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 7! = 5040.

Question 5

A word has 8 letters: 2 as, 3 bs, 3 cs. How many distinct ways can these letters be arranged?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 8
- Distribution: Type 1: 2, Type 2: 3, Type 3: 3

Step 1 - Total arrangements if all were distinct:
8! = 40320

Step 2 - Account for identical objects:
8! = 40320 / 2! = 2 / 3! = 6 / 3! = 6

Final Calculation:
= 560

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 8! = 40320.

Question 6

A word has 12 letters: 2 as, 2 bs, 2 cs, 3 ds, 3 es. How many distinct ways can these letters be arranged?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 12
- Distribution: Type 1: 2, Type 2: 2, Type 3: 2, Type 4: 3, Type 5: 3

Step 1 - Total arrangements if all were distinct:
12! = 479001600

Step 2 - Account for identical objects:
12! = 479001600 / 2! = 2 / 2! = 2 / 2! = 2 / 3! = 6 / 3! = 6

Final Calculation:
= 1663200

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 12! = 479001600.

Question 7

You have 13 books: 2 copies of book type A, 3 copies of book type B, 2 copies of book type C, 4 copies of book type D, 2 copies of book type E. How many distinct ways can these books be lined on a shelf?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 13
- Distribution: Type 1: 2, Type 2: 3, Type 3: 2, Type 4: 4, Type 5: 2

Step 1 - Total arrangements if all were distinct:
13! = 6227020800

Step 2 - Account for identical objects:
13! = 6227020800 / 2! = 2 / 3! = 6 / 2! = 2 / 4! = 24 / 2! = 2

Final Calculation:
= 5405400

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 13! = 6227020800.

Question 8

Consider 15 marbles: 4 of color 1, 3 of color 2, 2 of color 3, 3 of color 4, 3 of color 5. How many distinct ways can these marbles be arranged in a row?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 15
- Distribution: Type 1: 4, Type 2: 3, Type 3: 2, Type 4: 3, Type 5: 3

Step 1 - Total arrangements if all were distinct:
15! = 1307674368000

Step 2 - Account for identical objects:
15! = 1307674368000 / 4! = 24 / 3! = 6 / 2! = 2 / 3! = 6 / 3! = 6

Final Calculation:
= 126126000

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 15! = 1307674368000.

Question 9

You have 15 books: 3 copies of book type A, 4 copies of book type B, 2 copies of book type C, 4 copies of book type D, 2 copies of book type E. How many distinct ways can these books be lined on a shelf?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 15
- Distribution: Type 1: 3, Type 2: 4, Type 3: 2, Type 4: 4, Type 5: 2

Step 1 - Total arrangements if all were distinct:
15! = 1307674368000

Step 2 - Account for identical objects:
15! = 1307674368000 / 3! = 6 / 4! = 24 / 2! = 2 / 4! = 24 / 2! = 2

Final Calculation:
= 94594500

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 15! = 1307674368000.

Question 10

A word has 6 letters: 3 as, 2 bs, 1 cs. How many distinct ways can these letters be arranged?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 6
- Distribution: Type 1: 3, Type 2: 2, Type 3: 1

Step 1 - Total arrangements if all were distinct:
6! = 720

Step 2 - Account for identical objects:
6! = 720 / 3! = 6 / 2! = 2

Final Calculation:
= 60

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 6! = 720.

Question 11

A word has 8 letters: 4 as, 3 bs, 1 cs. How many distinct ways can these letters be arranged?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 8
- Distribution: Type 1: 4, Type 2: 3, Type 3: 1

Step 1 - Total arrangements if all were distinct:
8! = 40320

Step 2 - Account for identical objects:
8! = 40320 / 4! = 24 / 3! = 6

Final Calculation:
= 280

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 8! = 40320.

Question 12

You have 12 books: 3 copies of book type A, 2 copies of book type B, 2 copies of book type C, 2 copies of book type D, 3 copies of book type E. How many distinct ways can these books be lined on a shelf?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 12
- Distribution: Type 1: 3, Type 2: 2, Type 3: 2, Type 4: 2, Type 5: 3

Step 1 - Total arrangements if all were distinct:
12! = 479001600

Step 2 - Account for identical objects:
12! = 479001600 / 3! = 6 / 2! = 2 / 2! = 2 / 2! = 2 / 3! = 6

Final Calculation:
= 1663200

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 12! = 479001600.

Question 13

A word has 13 letters: 4 as, 4 bs, 2 cs, 3 ds. How many distinct ways can these letters be arranged?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 13
- Distribution: Type 1: 4, Type 2: 4, Type 3: 2, Type 4: 3

Step 1 - Total arrangements if all were distinct:
13! = 6227020800

Step 2 - Account for identical objects:
13! = 6227020800 / 4! = 24 / 4! = 24 / 2! = 2 / 3! = 6

Final Calculation:
= 900900

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 13! = 6227020800.

Question 14

Consider 11 marbles: 2 of color 1, 3 of color 2, 2 of color 3, 2 of color 4, 2 of color 5. How many distinct ways can these marbles be arranged in a row?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 11
- Distribution: Type 1: 2, Type 2: 3, Type 3: 2, Type 4: 2, Type 5: 2

Step 1 - Total arrangements if all were distinct:
11! = 39916800

Step 2 - Account for identical objects:
11! = 39916800 / 2! = 2 / 3! = 6 / 2! = 2 / 2! = 2 / 2! = 2

Final Calculation:
= 415800

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 11! = 39916800.

Question 15

You have 14 books: 4 copies of book type A, 2 copies of book type B, 3 copies of book type C, 4 copies of book type D, 1 copies of book type E. How many distinct ways can these books be lined on a shelf?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 14
- Distribution: Type 1: 4, Type 2: 2, Type 3: 3, Type 4: 4, Type 5: 1

Step 1 - Total arrangements if all were distinct:
14! = 87178291200

Step 2 - Account for identical objects:
14! = 87178291200 / 4! = 24 / 2! = 2 / 3! = 6 / 4! = 24

Final Calculation:
= 12612600

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 14! = 87178291200.

Question 16

A word has 12 letters: 4 as, 2 bs, 3 cs, 3 ds. How many distinct ways can these letters be arranged?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 12
- Distribution: Type 1: 4, Type 2: 2, Type 3: 3, Type 4: 3

Step 1 - Total arrangements if all were distinct:
12! = 479001600

Step 2 - Account for identical objects:
12! = 479001600 / 4! = 24 / 2! = 2 / 3! = 6 / 3! = 6

Final Calculation:
= 277200

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 12! = 479001600.

Question 17

You have 7 books: 4 copies of book type A, 2 copies of book type B, 1 copies of book type C. How many distinct ways can these books be lined on a shelf?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 7
- Distribution: Type 1: 4, Type 2: 2, Type 3: 1

Step 1 - Total arrangements if all were distinct:
7! = 5040

Step 2 - Account for identical objects:
7! = 5040 / 4! = 24 / 2! = 2

Final Calculation:
= 105

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 7! = 5040.

Question 18

You have 12 books: 4 copies of book type A, 4 copies of book type B, 3 copies of book type C, 1 copies of book type D. How many distinct ways can these books be lined on a shelf?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 12
- Distribution: Type 1: 4, Type 2: 4, Type 3: 3, Type 4: 1

Step 1 - Total arrangements if all were distinct:
12! = 479001600

Step 2 - Account for identical objects:
12! = 479001600 / 4! = 24 / 4! = 24 / 3! = 6

Final Calculation:
= 138600

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 12! = 479001600.

Question 19

A word has 8 letters: 2 as, 2 bs, 3 cs, 1 ds. How many distinct ways can these letters be arranged?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 8
- Distribution: Type 1: 2, Type 2: 2, Type 3: 3, Type 4: 1

Step 1 - Total arrangements if all were distinct:
8! = 40320

Step 2 - Account for identical objects:
8! = 40320 / 2! = 2 / 2! = 2 / 3! = 6

Final Calculation:
= 1680

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 8! = 40320.

Question 20

You have 9 books: 4 copies of book type A, 3 copies of book type B, 2 copies of book type C. How many distinct ways can these books be lined on a shelf?
Step-by-Step Solution:

Concept: Permutations with identical objects. When objects of the same type are indistinguishable, we divide by the factorial of each type's count.

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$
where $n$ is total objects and $n_i$ is the count of type $i$.

Given Data:
- Total objects: 9
- Distribution: Type 1: 4, Type 2: 3, Type 3: 2

Step 1 - Total arrangements if all were distinct:
9! = 362880

Step 2 - Account for identical objects:
9! = 362880 / 4! = 24 / 3! = 6 / 2! = 2

Final Calculation:
= 1260

Key Principle: Each group of identical objects overcounts by a factor of (count)!. Division corrects this.

Verification: The result is an integer and less than 9! = 362880.
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