Given:

Total number of officers = 4

Total number of jawans = 8

Total number of selection to be made is 6

**(i) **to include exactly one officer

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

= (^{4}C_{1}) × (^{8}C_{5})

By using the formula,

^{n}C_{r} = \( \dfrac{ n!}{r!(n – r)!}\)

**(ii) **to include at least one officer?

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)

= ^{12}C_{6} – ^{8}C_{6}

By using the formula,

^{n}C_{r} = \( \dfrac{ n!}{r!(n – r)!}\)

= (11×2×3×2×7) – (4×7)

= 924 – 28

= 896 ways

The required no. of ways are 224 and 896.

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