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Combinatorics: Non-adjacent Houses

XAT 2011 · Quantitative Ability & Data Interpretation · Easy · Permutations & Combinations

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In a locality, there are ten houses in a row. On a particular night a thief planned to steal from three houses of the locality. In how many ways can he plan such that no two of them are next to each other?

  1. A.

    56

  2. B.

    73

  3. C.

    80

  4. D.

    120

  5. E.

    None of the above

Answer

A

Explanation

Number of ways to choose kk non-adjacent items from nn items in a row is (n−k+1k)\binom{n - k + 1}{k}. Here n=10,k=3  ⟹  (10−3+13)=(83)=56n = 10, k = 3 \implies \binom{10 - 3 + 1}{3} = \binom{8}{3} = 56.

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