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GATE CS 2022 · Engineering Mathematics

Question 32

NAT1 markNo negative markingEasy (estimate)

Question

The number of arrangements of six identical balls in three identical bins is______.

Official answer and solution
Official answer
7
From
Official GATE 2022 answer key

Solution

List all integer partitions of 6 into at most 3 parts.

Placing 6 identical balls into 3 identical bins corresponds to finding the number of integer partitions of 6 into at most 3 parts, represented as triplets (x1,x2,x3)(x_1, x_2, x_3) such that x1≥x2≥x3≥0x_1 \ge x_2 \ge x_3 \ge 0 and x1+x2+x3=6x_1 + x_2 + x_3 = 6:

  1. (6,0,0)(6, 0, 0)
  2. (5,1,0)(5, 1, 0)
  3. (4,2,0)(4, 2, 0)
  4. (4,1,1)(4, 1, 1)
  5. (3,3,0)(3, 3, 0)
  6. (3,2,1)(3, 2, 1)
  7. (2,2,2)(2, 2, 2)

Enumerating all partitions yields exactly 7 distinct arrangements.

Final answer

7

Common trap

Using stars-and-bars (6+3−13−1)=(82)=28\binom{6 + 3 - 1}{3 - 1} = \binom{8}{2} = 28, which assumes distinct bins.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper
GATE 2022 Computer Science and Information Technology (CS)
Question
32 of 65
Section
Computer Science and Information Technology

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