# 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?

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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