Answer to Question #343052 in Discrete Mathematics for ferqo

Question

How many ways are there to select 12 countries in the United Nations to serve on a council if 2 is selected from a block of 55, 2 are selected from a block of 67 and 8 are selected from the remaining 67 countries

Answer 1

2 countries out of 55 can be selected in $C_{55}^2 = \frac{{55!}}{{2!(55 - 2)!}} = \frac{{2970}}{2} = 1485$ 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;

8 countries out of 67 can be selected in $C_{67}^8 = \frac{{67!}}{{8!\left( {67 - 8} \right)!}} = \frac{{67!}}{{8!59!}} = 6522361560$ ways

Then the answer is

$n = 1485\cdot 2211 \cdot 6522361560 = 21415097992602600$