In how many ways can ten people be chosen from a group of 14

1,001

Step-by-step explanation:

Given that the question asks only for 10 people and order doesn't matter, this can be identified as a combination question, using the formula nCr = n! / r! * (n - r) ! where n is the total number of whatever you're dealing with (14 people, in this case) and r is the number of items being chosen (10 people, for this problem). The exclamation marks indicate factorials (meaning the collective product of the number and every positive integer before it). Given this it would be set up as such:

nCr = 14! / 10! * (14 - 10) !

If you're lazy like me and use a calculator, you could just plug this in, but if you're also like me and occasionally do competitive math, you won't always be allowed to use your calculator, and this becomes much easier to solve manually if you expand the factorials.

nCr = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 / 1 * 2 * 3 ... * 10 * 1 * 2 * 3 * 4

From here you can simplify the problem significantly, canceling out a bunch of the factors.

nCr = 11 * 12 * 13 * 14 / 1 * 2 * 3 * 4

Multiply.

nCr = 24,024 / 24

Divide.

1,001