Question

How many ways are there to select 9 countries in the United Nations to serve on a council if 1 is selected from a block of 50, 2 are selected from a block of 67 and 6 are selected from the remaining 72 countries?

Answer 1

1 country out of 50 can be selected in 50 ways;

2 countrues out of 67 can be selected in $C_{67}^2 = \frac{{67!}}{{2!(67 - 2)!}} = \frac{{66 \cdot 67}}{2} = 2211$ ways;

6 countries out of 72 can be selected in $C_{72}^6 = \frac{{72!}}{{6!\left( {72 - 6} \right)!}} = \frac{{72!}}{{6!66!}} = 156238908$ ways

Then the answer is

$n = 50 \cdot 2211 \cdot 156238908 = 17272211279400$