SHOW WORK

If a person can select 7 cups from 10 cups at a coffee shop, how many different combinations are there?

There are 604800 ways of selecting 7 cups from 10 cups.

Solution:

Let's for sake of understanding, we simplify our question.

Let's say there are 3 cups A, B, C. In how many ways two cups can be selected.

So for selecting first cup we have three option that are A, B or C.

And once first cup is selected, number of options for selecting second cup is two.

So it is found that number of selecting 2 cups from 3 cups is product of option available for selecting first cup multiplied by option available for selecting second cup = 3 x 2 = 6

So now applying same logic for selecting 7 cups from 10 cups we get

For selecting first cup option available = 10 cups

Once first cup is selected option available to select second cup = 9 cups

Once second cup is also selected option available to select third cup = 8 cups

Once third cup is also selected option available to select fourth cup = 7 cups

Once fourth cup is also selected option available to select fifth cup = 6 cups

Once fifth cup is also selected option available to select sixth cup = 5 cups

Once sixth cup is also selected option available to select seventh cup = 4 cups

So number of selecting 7 cups from 10 cups = 10 x 9 x 8 x 7 x 6 x 5 x 4 = 604800.

Hence there are 604800 ways of selecting 7 cups from 10 cups.