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ABHYAS PREP: ADVANCED QUANT · PERMUTATION & COMBINATION

Selections and committees with conditions

Choosing (order does not matter) is nCr. Split conditions into cases and multiply within a case; for “at least one”, subtract the “none” case from the total.

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Worked examples

Example 1 · level 1

At Vihaan’s alumni meet in Mysuru, each of the 14 people present shakes hands with every other person exactly once. How many handshakes take place?

  1. 78
  2. 91
  3. 182
  4. 196
Show solution
  1. Each handshake is one pair: C(14, 2).
  2. = 14 × 13 ÷ 2 = 91.

Answer: 91

Example 2 · level 2

A team of 3 is to be chosen from 12 people for a housing society in Ranchi. In how many ways can it be chosen if Diya, one of the 12, must be left out?

  1. 55
  2. 165
  3. 219
  4. 220
Show solution
  1. Diya is out, so choose 3 from 11.
  2. C(11, 3) = 165.

Answer: 165

Example 3 · level 3

From 8 men and 4 women, a team of 3 is to be formed for a housing society in Indore. In how many ways can it be formed if it must include at least one woman?

  1. 112
  2. 164
  3. 216
  4. 220
Show solution
  1. All committees: C(12, 3) = 220.
  2. No woman (all men): C(8, 3) = 56.
  3. At least one woman = 220 − 56 = 164.

Answer: 164

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