Question

In how many ways can $6$ persons be selected from $4$ officers and $8$ constables, if at least one officer is to be?

• A. $224$
• B. $672$
• C. $896$
• D. None of these

Answer 1

The correct option is C

$896$

Explanation for the correct option:

Finding the number of ways to select $6$ persons such that at least one officer is included:

Number of persons to be selected = $6$

Number of officers = $4$

Number of constables = $8$

The total number of ways to select $6$ persons such that at least one officer must be included are as follows

Possibility $1$: one officer out of $4$ and $5$constables out of $8$$={}^4{C}_1\times {}^8{C}_5=224$

Possibility $2$: $2$ officers out of $4$ and $4$ constables out of $8$ $={}^4{C}_2\times {}^8{C}_4=420$

Possibility $3$: $3$ officers out of $4$ and $3$ constables out of $8$ $={}^4{C}_3\times {}^8{C}_3=224$

Possibility $4$: $4$ officers out of $4$ and $2$ constables out of $8$ $={}^4{C}_4\times {}^8{C}_2=28$

Thus the required number of ways to select the persons $$\begin{gathered} =224+420+224+28 \\ =896 \end{gathered}$$

Hence, option (C) is correct.