### Video Transcript

Let set 𝑋 contain the values 𝑥, where 𝑥 is an integer that is greater than or equal to seven and less than or equal to 16 and 𝑌 be the ordered pairs 𝑎, 𝑏, where 𝑎 and 𝑏 exist in set 𝑥 and 𝑎 is not equal to 𝑏. Determine the value of 𝑛 of 𝑦, where 𝑛 of 𝑦 is the number of elements in 𝑌.

We are told in the question that 𝑥 is the set of integers greater than or equal to seven and less than or equal to 16. This means that 𝑥 is the integers between seven and 16 inclusive, a total of 10 elements. Set 𝑌 is the set of ordered pairs of these elements with no repetition. One way of solving this problem would be using the fundamental counting principle.

As there are 10 elements of set 𝑋, there are 10 possible values of 𝑎. Since 𝑎 is not equal to 𝑏, there are then nine possible values of 𝑏. Multiplying 10 by nine gives us an answer of 90. This means that there are a total of 90 possible elements in 𝑌. There are 90 different ordered pairs from the set of integers from seven to 16.

Alternatively, we could’ve used our knowledge of permutations without repetition. Here, 𝑛P𝑟 is equal to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial. There are 10 elements in set 𝑋. And we are selecting two of them for each ordered pair. This means that we need to calculate 10P two. This is equal to 10 factorial divided by eight factorial. The numerator can be rewritten as 10 multiplied by nine multiplied by eight factorial. We can then divide the numerator and denominator by eight factorial, once again leaving us with 10 multiplied by nine. This confirms that there are 90 ordered pairs in set 𝑌.