Partition into Unequal Groups - Expert Level: conceptual clarity Partition into Unequal Groups EXPERT

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 10
- Group sizes: 7, 2, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 7 people from 10: C(10, 7) = 120

Step 2 - Choose second group:
From remaining 3 people, choose 2: C(3, 2) = 3

Continue for all groups:
C(10,7) = 120
C(3,2) = 3
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 120 × 3 × 1 (last group)
= 360

Simplified formula:
= 10! / (7! × 2! × 1!)
= 3628800 / (5040 × 2 × 1)
= 360

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 10 = 10 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 10
- Group sizes: 7, 2, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 7 people from 10: C(10, 7) = 120

Step 2 - Choose second group:
From remaining 3 people, choose 2: C(3, 2) = 3

Continue for all groups:
C(10,7) = 120
C(3,2) = 3
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 120 × 3 × 1 (last group)
= 360

Simplified formula:
= 10! / (7! × 2! × 1!)
= 3628800 / (5040 × 2 × 1)
= 360

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 10 = 10 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 10
- Group sizes: 5, 4, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 5 people from 10: C(10, 5) = 252

Step 2 - Choose second group:
From remaining 5 people, choose 4: C(5, 4) = 5

Continue for all groups:
C(10,5) = 252
C(5,4) = 5
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 252 × 5 × 1 (last group)
= 1260

Simplified formula:
= 10! / (5! × 4! × 1!)
= 3628800 / (120 × 24 × 1)
= 1260

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 10 = 10 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 12
- Group sizes: 8, 3, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 8 people from 12: C(12, 8) = 495

Step 2 - Choose second group:
From remaining 4 people, choose 3: C(4, 3) = 4

Continue for all groups:
C(12,8) = 495
C(4,3) = 4
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 495 × 4 × 1 (last group)
= 1980

Simplified formula:
= 12! / (8! × 3! × 1!)
= 479001600 / (40320 × 6 × 1)
= 1980

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 12 = 12 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 10
- Group sizes: 7, 2, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 7 people from 10: C(10, 7) = 120

Step 2 - Choose second group:
From remaining 3 people, choose 2: C(3, 2) = 3

Continue for all groups:
C(10,7) = 120
C(3,2) = 3
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 120 × 3 × 1 (last group)
= 360

Simplified formula:
= 10! / (7! × 2! × 1!)
= 3628800 / (5040 × 2 × 1)
= 360

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 10 = 10 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 12
- Group sizes: 7, 4, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 7 people from 12: C(12, 7) = 792

Step 2 - Choose second group:
From remaining 5 people, choose 4: C(5, 4) = 5

Continue for all groups:
C(12,7) = 792
C(5,4) = 5
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 792 × 5 × 1 (last group)
= 3960

Simplified formula:
= 12! / (7! × 4! × 1!)
= 479001600 / (5040 × 24 × 1)
= 3960

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 12 = 12 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 8
- Group sizes: 5, 2, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 5 people from 8: C(8, 5) = 56

Step 2 - Choose second group:
From remaining 3 people, choose 2: C(3, 2) = 3

Continue for all groups:
C(8,5) = 56
C(3,2) = 3
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 56 × 3 × 1 (last group)
= 168

Simplified formula:
= 8! / (5! × 2! × 1!)
= 40320 / (120 × 2 × 1)
= 168

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 8 = 8 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 10
- Group sizes: 6, 3, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 6 people from 10: C(10, 6) = 210

Step 2 - Choose second group:
From remaining 4 people, choose 3: C(4, 3) = 4

Continue for all groups:
C(10,6) = 210
C(4,3) = 4
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 210 × 4 × 1 (last group)
= 840

Simplified formula:
= 10! / (6! × 3! × 1!)
= 3628800 / (720 × 6 × 1)
= 840

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 10 = 10 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 12
- Group sizes: 6, 5, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 6 people from 12: C(12, 6) = 924

Step 2 - Choose second group:
From remaining 6 people, choose 5: C(6, 5) = 6

Continue for all groups:
C(12,6) = 924
C(6,5) = 6
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 924 × 6 × 1 (last group)
= 5544

Simplified formula:
= 12! / (6! × 5! × 1!)
= 479001600 / (720 × 120 × 1)
= 5544

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 12 = 12 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 9
- Group sizes: 8, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 8 people from 9: C(9, 8) = 9

Step 2 - Choose second group:
From remaining 1 people, choose 1: C(1, 1) = 1

Continue for all groups:
C(9,8) = 9
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 9 × 1 (last group)
= 9

Simplified formula:
= 9! / (8! × 1!)
= 362880 / (40320 × 1)
= 9

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 9 = 9 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 11
- Group sizes: 6, 4, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 6 people from 11: C(11, 6) = 462

Step 2 - Choose second group:
From remaining 5 people, choose 4: C(5, 4) = 5

Continue for all groups:
C(11,6) = 462
C(5,4) = 5
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 462 × 5 × 1 (last group)
= 2310

Simplified formula:
= 11! / (6! × 4! × 1!)
= 39916800 / (720 × 24 × 1)
= 2310

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 11 = 11 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 12
- Group sizes: 7, 4, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 7 people from 12: C(12, 7) = 792

Step 2 - Choose second group:
From remaining 5 people, choose 4: C(5, 4) = 5

Continue for all groups:
C(12,7) = 792
C(5,4) = 5
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 792 × 5 × 1 (last group)
= 3960

Simplified formula:
= 12! / (7! × 4! × 1!)
= 479001600 / (5040 × 24 × 1)
= 3960

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 12 = 12 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 10
- Group sizes: 5, 3, 2
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 5 people from 10: C(10, 5) = 252

Step 2 - Choose second group:
From remaining 5 people, choose 3: C(5, 3) = 10

Continue for all groups:
C(10,5) = 252
C(5,3) = 10
C(2,2) = 1 (last group)

Step 3 - Multiply:
Total ways = 252 × 10 × 1 (last group)
= 2520

Simplified formula:
= 10! / (5! × 3! × 2!)
= 3628800 / (120 × 6 × 2)
= 2520

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 10 = 10 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 11
- Group sizes: 8, 2, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 8 people from 11: C(11, 8) = 165

Step 2 - Choose second group:
From remaining 3 people, choose 2: C(3, 2) = 3

Continue for all groups:
C(11,8) = 165
C(3,2) = 3
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 165 × 3 × 1 (last group)
= 495

Simplified formula:
= 11! / (8! × 2! × 1!)
= 39916800 / (40320 × 2 × 1)
= 495

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 11 = 11 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 8
- Group sizes: 5, 2, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 5 people from 8: C(8, 5) = 56

Step 2 - Choose second group:
From remaining 3 people, choose 2: C(3, 2) = 3

Continue for all groups:
C(8,5) = 56
C(3,2) = 3
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 56 × 3 × 1 (last group)
= 168

Simplified formula:
= 8! / (5! × 2! × 1!)
= 40320 / (120 × 2 × 1)
= 168

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 8 = 8 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 10
- Group sizes: 5, 4, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 5 people from 10: C(10, 5) = 252

Step 2 - Choose second group:
From remaining 5 people, choose 4: C(5, 4) = 5

Continue for all groups:
C(10,5) = 252
C(5,4) = 5
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 252 × 5 × 1 (last group)
= 1260

Simplified formula:
= 10! / (5! × 4! × 1!)
= 3628800 / (120 × 24 × 1)
= 1260

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 10 = 10 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 10
- Group sizes: 7, 2, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 7 people from 10: C(10, 7) = 120

Step 2 - Choose second group:
From remaining 3 people, choose 2: C(3, 2) = 3

Continue for all groups:
C(10,7) = 120
C(3,2) = 3
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 120 × 3 × 1 (last group)
= 360

Simplified formula:
= 10! / (7! × 2! × 1!)
= 3628800 / (5040 × 2 × 1)
= 360

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 10 = 10 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 9
- Group sizes: 5, 3, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 5 people from 9: C(9, 5) = 126

Step 2 - Choose second group:
From remaining 4 people, choose 3: C(4, 3) = 4

Continue for all groups:
C(9,5) = 126
C(4,3) = 4
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 126 × 4 × 1 (last group)
= 504

Simplified formula:
= 9! / (5! × 3! × 1!)
= 362880 / (120 × 6 × 1)
= 504

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 9 = 9 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 7
- Group sizes: 4, 2, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 4 people from 7: C(7, 4) = 35

Step 2 - Choose second group:
From remaining 3 people, choose 2: C(3, 2) = 3

Continue for all groups:
C(7,4) = 35
C(3,2) = 3
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 35 × 3 × 1 (last group)
= 105

Simplified formula:
= 7! / (4! × 2! × 1!)
= 5040 / (24 × 2 × 1)
= 105

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 7 = 7 ✓

Step-by-Step Solution:

Concept: Partitioning into groups of specified sizes. When group sizes are different, groups are automatically distinguishable by size.

Given:
- Total people: 10
- Group sizes: 7, 2, 1
- Groups are unlabeled (no names like Team A, Team B)

Formula:
$$\frac{n!}{n_1! \cdot n_2! \cdot ... \cdot n_k!}$$

Step 1 - Choose first group:
Choose 7 people from 10: C(10, 7) = 120

Step 2 - Choose second group:
From remaining 3 people, choose 2: C(3, 2) = 3

Continue for all groups:
C(10,7) = 120
C(3,2) = 3
C(1,1) = 1 (last group)

Step 3 - Multiply:
Total ways = 120 × 3 × 1 (last group)
= 360

Simplified formula:
= 10! / (7! × 2! × 1!)
= 3628800 / (5040 × 2 × 1)
= 360

Key Insight: Since groups have different sizes, we don't divide by k! (they're naturally distinguishable by their sizes).

Contrast with equal groups:
- If groups were same size: would divide by k!
- Here, sizes differ: no division needed

Verification: Sum of group sizes = 10 = 10 ✓