Question 29 of 42intermediate🔧 ApplyNumerical3 marks

In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?

Correct Answer

3960 ways

Exercise: EXERCISE 6.4 | Q: 7 | (Page 20)
For More Understanding

Explanation

This is a combination problem where we need to select a cricket team with specific constraints. The team must have exactly 4 bowlers from the available 5 bowlers, and the remaining 7 players must be selected from the 12 non-bowlers. We use the multiplication principle since both selections happen simultaneously and independently.

Solution Steps

  1. Step 1: Identify the given information. Total players = 17 Number of bowlers = 5 Number of non-bowlers = 17 - 5 = 12 Team size = 11 players Required bowlers in team = 4

  2. Step 2: Determine the number of ways to select bowlers. We need to select exactly 4 bowlers from 5 bowlers. Number of ways = ⁵C₄ = 5!/(4! × 1!) = 5

  3. Step 3: Determine the number of ways to select non-bowlers. We need to select 7 non-bowlers from 12 non-bowlers (since 11 - 4 = 7). Number of ways = ¹²C₇ = 12!/(7! × 5!) = (12×1112 \times 11 × 10×910 \times 9 × 8)/(5×45 \times 4 × 3×23 \times 2 × 1) = 95040/120 = 792

  4. Step 4: Calculate the total number of ways. Total number of ways = ⁵C₄ × ¹²C₇ = 5×7925 \times 792 = 3960 ways

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