# Video: Applications of the Counting Principle that Involve Replacement and No Order

Three friends are in a tapas restaurant. They would like to order 6 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 6 dishes to share?

02:50

### 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 we’re counting with or without replacement and whether or not order matters. In this question, we’re 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 𝑛 plus 𝑟 minus one factorial divided by 𝑟 factorial multiplied by 𝑛 minus one factorial. There are a total of 15 different options on the menu. Therefore, 𝑛 is equal to 15. The friends would like to order six dishes to share. Therefore, 𝑟 is equal to six.

Substituting in these values gives us a numerator of 15 plus six minus one facorial. 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 𝑥 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.