Question

How many ways are there to select 12 countries in the United Nations to serve on a council if 3 is selected from a block of 53, 1 are selected from a block of 62 and 8 are selected from the remaining 74 countries?

Answer 1

By combinatorial product principle, the number of ways is equal to

$C_{53}^3\cdot C_{62}^1\cdot C_{74}^8 =\frac{53\cdot 52\cdot 51}{2\cdot 3}\cdot62\cdot \frac{74\cdot 73\cdot 72\cdot 71\cdot 70\cdot 69\cdot 68\cdot 67}{2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8}\approx 2.189\cdot 10^{16}.$