There are p=5 women and q=9 men.
6 people have to be chosen and out of it more than 3 must be men.
So, the people can be chosen as:
• 4 men and 2women
or
• 5 men and 1 woman
or
• 6 men and no woman.
So, the number of ways of choosing 6 people if more than 3 must be men is,
[tex]\begin{gathered} N=n(4\text{ men and 2 woman)+n(5 men and 1 woman)+n(6 men)} \\ =^qC_4\times^pC_2+^qC_5\times^pC_1+^qC_6 \\ =^9C_4\times^5C_2+^9C_5\times^5C_1+^9C_6 \\ =126\times10+126\times5+84 \\ =1260+630+84 \\ =1974 \end{gathered}[/tex]Therefore, there are 1974 ways of choosing 6 people if more than 3 must be men.
