Five numbers x₁,x₂,x₃,x₄,x₅ are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order (x₁ < x₂ < x₃ < x₄ < x₅). The probability that x₂ = 7 and x₄ = 11 is :

Solution

Maths sir · 3 steps

Pin x₂ = 7 and x₄ = 11, then fill each gap around them independently.

1. Total selections. Any 5 different numbers from 18 can be arranged in increasing order in exactly one way. So the total number of outcomes is C(18, 5) = 8568.
2. Fill the gaps. x₁ < 7 must come from {1, …, 6}, which gives 6 choices. 7 < x₃ < 11 must come from {8, 9, 10}, which gives 3 choices. x₅ > 11 must come from {12, …, 18}, which gives 7 choices.

   P = (6 × 3 × 7)/C(18, 5) = 126/8568

3. So. Since 8568 = 126 × 68, the probability is 1/68, option (c).

The idea to keep. Once some positions in an increasing arrangement are fixed, each gap between them is filled on its own. Multiply the number of choices for each gap.

Answer from the NTA final answer key. Solution by PrepWiser.