A sports delegation of 6 athletes is to be selected from a pool of 7 sprinters and 5 dista… — JAMB UTME

A sports delegation of $6$ athletes is to be selected from a pool of $7$ sprinters and $5$ distance runners. If the delegation must contain at least $4$ sprinters, in how many different ways can the delegation be formed?

Answer: 462 ways

Answer

The total number of different ways to form the delegation is 462.

To select at least 4 sprinters in a delegation of 6 athletes, we must evaluate three mutually exclusive cases: selecting 4 sprinters and 2 distance runners (

{}^7\text{C}_4 \times {}^5\text{C}_2 = 350

), selecting 5 sprinters and 1 distance runner (

{}^7\text{C}_5 \times {}^5\text{C}_1 = 105

), and selecting 6 sprinters and 0 distance runners (

{}^7\text{C}_6 \times {}^5\text{C}_0 = 7

). Adding these yields

350 + 105 + 7 = 462

total ways.

Step-by-Step Solution

Determine all valid combinations of sprinters and distance runners satisfying the condition of having at least 4 sprinters in a group of 6.

Three valid cases: (4 sprinters, 2 distance runners), (5 sprinters, 1 distance runner), and (6 sprinters, 0 distance runners).

The delegation requires 6 members and at least 4 sprinters.

Calculate the combinations for each case using

{}^n\text{C}_r = \frac{n!}{r!(n-r)!}

Case 1:

{}^7\text{C}_4 \times {}^5\text{C}_2 = 350

; Case 2:

{}^7\text{C}_5 \times {}^5\text{C}_1 = 105

; Case 3:

{}^7\text{C}_6 \times {}^5\text{C}_0 = 7

Apply the product rule of counting for selecting sprinters and distance runners independently within each case.

Sum the results of the mutually exclusive cases.

350 + 105 + 7 = 462.

Apply the addition principle of counting for mutually exclusive scenarios.

Key Concept

Combinations with restrictions and the addition principle of counting

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic