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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
$40$
$41$
$16$
$32$
Solution
$\left| {\begin{array}{*{20}{c}}
L&O&Y \\
2&2&4 \\
{ \geqslant 1}&{ \geqslant 1}&{2 \leqslant }
\end{array}} \right| \Rightarrow \left| {\begin{array}{*{20}{c}}
L&O&Y \\
1&1&2 \\
1&2&1 \\
2&1&1 \\
2&2&0
\end{array}} \right|$
Required number of ways
${ = ^2}{C_1}{ \times ^2}{C_1}{ \times ^2}{C_2}{ + ^2}{C_1}{ \times ^2}{C_2}{ \times ^4}{C_1}$ ${ + ^2}{C_2}{ \times ^2}{C_1}{ \times ^4}{C_1}{ + ^2}{C_2}{ \times ^2}{C_2}{ \times ^4}{C_0}$
$ = 2 \times 2 \times \frac{{4 \times 3}}{2}$ $ + 2 \times 1 \times 4 + 1 \times 2 \times 4 + 1 \times 1 \times 1$
$ = 24 + 8 + 8 + 1 = 41$
