Multiple choice

$10$ distinct balls are arranged in a row. The number of ways of selecting three of these balls so that no two of them are next to each other is

  1. $\dfrac {1}{6}\times 8\times 7\times 6$
  2. $^{8}C_{3}$
  3. $^{7}C_{3} + ^{7}C_{2}$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To select 3 balls from 10 such that no two are adjacent, we use the formula (n-k+1)C(k), where n=10, k=3. This is 8C3 = (8*7*6)/(3*2*1) = 56. Option A is (1/6)*8*7*6 = 56.

AI explanation

To select 3 non-adjacent balls from a row of 10, we use the gap method, which requires choosing 3 positions from the 8 available gaps created around the remaining 7 balls. The number of ways is therefore 8C3. Evaluating this combination gives (8 * 7 * 6) / (3 * 2 * 1), which is 1/6 * 8 * 7 * 6.

Keep practicing — related questions