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 = 12C68C6


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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