Question

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 1

Total number of officer = 4

Total number of jawans = 8

Total number of selection to be made = 6

(i) To include exactly one officer

This can be done is ${}^4{C}_1{\times }^8{C}_5$ ways

$=\frac{4!}{1!3!}\times \frac{8!}{5!3!}$ = 224 ways

$=\frac{4\times 8\times 7\times 6}{3\times 2}=224$ ways

(ii) To include at least one officer

This can be done is following ways

${}^4{C}_1{\times }^8{C}_5{+}^8{C}_4{+}^4{C}_3{\times }^8{C}_3{+}^4{C}_4{\times }^8{C}_2$

$=\frac{4\times 8!}{5!3!}+\frac{4!}{2!2!}\times \frac{8!}{4!4!}\times \frac{4!}{3!1!}\times \frac{8!}{3!5!}+\frac{1\times 8!}{2!6!}$

$=\left(\frac{4\times 8\times 7\times 6}{3\times 2}\right)+\left(\frac{4\times 3\times 8\times 7\times 6\times 5}{2\times 4\times 3\times 2}\right)+\left(\frac{4\times 8\times 7\times 6}{3\times 2}\right)+\left(\frac{8\times 7}{2\times 1}\right)=(4\times 8\times 7)+(4\times 3\times 7\times 5)+(4\times 8\times 7)+(4\times 7)=224+420+224+28$
= 896 ways