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Official PYQ · 1990Q3-ixCombinatorics: Counting, Recurrence Relations, Generating FunctionsMCQ1 mark

GATE CS 1990 Q3-ix — Combinatorics: Counting, Recurrence Relations, Generating Functions

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The number of ways in which 5  A′s,5  B′s5\; A's, 5\; B's and 5  C′s5\; C's can be arranged in a row is:

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Answer: A

We need to find the number of ways to arrange 1515 letters in a row, where there are:

  • 55 identical AA's
  • 55 identical BB's
  • 55 identical CC's

The total number of objects to arrange is n=5+5+5=15n = 5 + 5 + 5 = 15.

Using the formula for the permutations of a multiset (where some objects are identical):

Number of arrangements=n!nA!⋅nB!⋅nC!\text{Number of arrangements} = \frac{n!}{n_A! \cdot n_B! \cdot n_C!}

Substituting the given values:

Number of arrangements=15!5!⋅5!⋅5!=15!(5!)3\text{Number of arrangements} = \frac{15!}{5! \cdot 5! \cdot 5!} = \frac{15!}{(5!)^3}

This matches Option A.

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