Question

There are three engineers, three IAS officers and one both IAS and engineer. A committee of 3 is to be formed at random. The probability that at least one engineer and at least one IAS officer is in the committee is

• A. $\frac{2}{35}$
• B. $\frac{33}{35}$
• C. $\frac{3}{7}$
• D. $\frac{3}{35}$

Answer 1

Answer: (B)

$\frac{33}{35}$

Let $A$ be the set of $3$ engineers
Let $B$ be the set of $3$ $IAS$ officers
Let $C$ be the set of $1$ engineer-cum-IAS officer
The required selection may be done as follows:
(i) $1$ member from $A$ and $2$ members from $B$
(ii) $2$ members from $A$ and $1$ member from $B$
(iii) $1$ member from $A,1$ member from $B$ and $1$ member from $C$
(iv) $2$ members from $A$ and $1$ member from $C$
(v) $2$ members from $B$ and $1$ member from $C$
Thus, the required probability = $\frac{{(}_1^{3}C{\ast }_2^{3}C)+{(}_2^{3}C{\ast }_1^{3}C)+{(}_1^{3}C{\ast }_1^{3}C{\ast }_1^{1}C)+{(}_2^{3}C{\ast }_1^{1}C)+{(}_2^{3}C{\ast }_1^{1}C)}{{}_3^{7}C}=\frac{33}{35}$