Question

A group of 45 civil servants at the State Department are newly qualified to serve as Ambassadors to foreign governments. There are 22 countries that currently need Ambassadors. How many distinct groups of 22 people can the President promote to fill these jobs?A group of 45 civil servants at the State Department are
newly qualified to serve as Ambassadors to foreign governments. There are 22
countries that currently need Ambassadors. How many distinct groups of 22
people can the President promote to fill these jobs?

Answer 1

Answer: $4.1167 X 10^{12}$

Step-by-step explanation: You can search for 'binomial coefficient', or 'n choose k', and you should find the following:

$C_{k}(n) = ( \frac{n}{k}) = \frac{n!}{k!(n - k)!}$

It basically says: "From a group of 'n', how many unique groups of 'k' can I make?"

Or: "From a group of 45 civil servants, how many distinct groups of 22 people can I make?

So we have:

$n = 45$

$k = 22$

$C_{22}({45) = \frac{45!}{22!(45 - 22)!} = \frac{45!}{22!23!} = 4.1167X10^{12}$