Q. 93.7( 3 Votes )
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 :
Given that we have 4 officers and 8 jawans, we need to choose 6 persons with the following conditions,
i. To include exactly one officer:
ii. To include at least one officer.
(i) It is told that we need to choose 6 persons with exactly one officer.
Let us assume the no. of ways of choosing to be N.
⇒ N = (no. of ways of choosing 1 officer and 5 jawans from 4 officers and 8 jawans)
⇒ N = (no. of ways of choosing 1 officer from 4 officers) × (no. of ways of choosing 5 jawans from 8 jawans)
⇒ N = (4C1) × (8C5)
We know that 
,
And also n! = (n)(n – 1)......2.1
⇒ 
⇒ 
⇒ 
⇒ N = 4 × 56
⇒ N = 224 ways.
(ii) It is told we need to choose 6 persons with at least 1 officers.
Let us assume the total no. of ways be N1
⇒ N1 = (No. of ways of choosing 6 persons with at least one officer)
⇒ N1 = (total no. of ways of choosing 6 persons from all 12 persons) – (no. of ways of choosing 6 persons without any officer)
⇒ N1 = 12C6 – 8C6
We know that 
,
And also n! = (n)(n – 1)......2.1
⇒ 
⇒ 
⇒ 
⇒ N1 = 924 – 28
⇒ N1 = 896 ways
∴ The required no. of ways are 224 and 896.
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