Question

A committee of $4$ persons is to be formed from $2$ ladies, $2$ old men and $4$young men such that it includes at least $1$ lady, at least $1$ old man and at most $2$ young men. Then the total number of ways in which this committee can be formed is :

• A. $40$
• B. $41$
• C. $16$
• D. $32$

Answer 1

The correct option is B

$41$

An explanation for the correct answer:

Step 1: Find all possible numbers of ways when there is, 1 lady + 1 old man + 2 young men.

We have been given that, a committee of $4$ persons is to be formed from $2$ ladies, $2$ old men and $4$young men such that it includes at least $1$ lady, at least $1$ old man and at most $2$ young men.

We need to find the total number of ways in which this committee can be formed.

A possible number of ways when there is, $1$lady + $1$ old man + $2$ young men would be,

$$\begin{gathered} ={}^{2}C_1\times {}^{2}C_1\times {}^{4}C_2 \\ =\frac{2!}{1!(2-1)!}\times \frac{2!}{1!(2-1)!}\times \frac{4!}{2!(4-2)!} \\ =2\times 2\times 6 \\ =24 \end{gathered}$$

Step 2: Find all possible ways when there are, $\mathbf{2}$ ladies + $\mathbf{1}$ old man + $\mathbf{1}$ young man

$$\begin{gathered} ={}^{2}C_2\times {}^{2}C_1\times {}^{4}C_1 \\ =\frac{2!}{2!(2-2)!}\times \frac{2!}{1!(2-1)!}\times \frac{4!}{1!(4-1)!} \\ =1\times 2\times 4 \\ =8 \end{gathered}$$

Step 3: Find the possible number of ways when there are, $\mathbf{2}$ ladies + $\mathbf{2}$ old men + $\mathbf{0}$young man

$$\begin{gathered} ={}^{2}C_2\times {}^{2}C_2 \\ =\frac{2!}{2!(2-2)!}\times \frac{2!}{2!(2-2)!} \\ =1\times 1 \\ =1 \end{gathered}$$

Step 4: Find the possible number of ways when there is, $\mathbf{1}$ lady + $\mathbf{2}$ old men + $\mathbf{1}$ young man

$$\begin{gathered} ={}^{2}C_1\times {}^{2}C_2\times {}^{4}C_1 \\ =\frac{2!}{1!(2-1)!}\times \frac{2!}{2!(2-2)!}\times \frac{4!}{1!(4-1)!} \\ =2\times 1\times 4 \\ =8 \end{gathered}$$

Step 5: Find the total number of ways in which this committee can be formed

$$\begin{gathered} =24+8+1+8 \\ =41 \end{gathered}$$

Therefore, option (B) is the correct answer.