Question

From 4 officers and 8 jawans in how many ways can 6 be chosen 

(i) to include exactly one officer 

(ii) to include at least one officer?

Answer 1

Given as

The total number of officers = 4

The total number of jawans = 8

The total number of selection to be made is 6

(i) to include exactly one officer

The number of ways = (no. of ways of choosing 1 officer from 4 officers) × (no. of ways of choosing 5 jawans from 8 jawans)

= (⁴C₁) × (⁸C₅)

On using the formula,

ⁿC_r = n!/r!(n – r)!

(ii) to include at least one officer

The number of ways = (total no. of ways of choosing 6 persons from all 12 persons) – (no. of ways of choosing 6 persons without any officer)

= ¹²C₆ – ⁸C₆

On using the formula,

ⁿC_r = n!/r!(n – r)!

= (11 × 2 × 3 × 2 × 7) – (4 × 7)

= 924 – 28

= 896 ways

Hence, the required no. of ways are 224 and 896.