### Video Transcript

Three friends are in a tapas
restaurant. They would like to order six dishes
to share. There are 15 different options on
the menu. Given that they can choose multiple
dishes of the same type, how many different possible ways can they order six dishes
to share?

When trying to solve any problem
involving combinations, we need to identify whether weβre counting with or without
replacement and whether or not order matters. In this question, weβre dealing
with repetition, and the order does not matter. This is because the three friends
can choose multiple dishes in any order. We can, therefore, use the formula
π plus π minus one factorial divided by π factorial multiplied by π minus one
factorial. There are a total of 15 different
options on the menu. Therefore, π is equal to 15. The friends would like to order six
dishes to share. Therefore, π is equal to six.

Substituting in these values gives
us a numerator of 15 plus six minus one factorial. The denominator is six factorial
multiplied by 15 minus one factorial. Simplifying the numerator gives us
20 factorial. And the denominator becomes six
factorial multiplied by 14 factorial. The factorial of any number is the
product of that integer and all smaller positive integers. This means that six factorial is
the product of six, five, four, three, two, and one. Multiplying these six numbers gives
us 720. Therefore, six factorial is equal
to 720.

Repeating this process for 14
factorial and 20 factorial would be very time-consuming. We can, therefore, use the
factorial button on the calculator. This is denoted by π₯ and then an
exclamation mark. Typing 20 factorial divided by six
factorial multiplied by 14 factorial into our calculator gives us 38760. We can, therefore, conclude that
there are 38760 possible ways that the three friends can order six dishes to
share.