# How many subsets of four elements can be formed from a set of 100 elements?

Dec 12, 2017

""^100C_4 = 3921225
If we choose the four elements one at a time, then there are $100$ possible choices for the $1$st element, $99$ for the $2$nd, $98$ for the $3$rd and $97$ for the $4$th. Note that it does not matter what order the four items are chosen in, so we then need to divide the total number of choices $100 \cdot 99 \cdot 98 \cdot 97$ by the number of ways of arranging $4$ items, namely 4! = 4 * 3 * 2 * 1.
The number of ways of choosing a subset of $4$ elements out of $100$ is:
""^100C_4 = (100!)/(96! 4!) = (100 * 99 * 98 * 97) / (4 * 3 * 2 * 1) = 3921225