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.