Video Transcript
Let uppercase 𝑋 be the set
containing 𝑥, where 𝑥 is an integer greater than or equal to 10 and less than or
equal to 16, and 𝑦 is the set containing 𝑎 and 𝑏, where 𝑎 and 𝑏 are elements of
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.
Since we’re looking to find the
number of elements in set 𝑦, let’s begin by inspecting set 𝑦 in a little more
detail. Firstly, we see we’re interested in
the two elements 𝑎 and 𝑏, which are themselves an element of set 𝑋. Then we also see that set 𝑋
contains only integers which are greater than or equal to 10 and less than or equal
to 16. In other words, set 𝑋 contains the
elements 10, 11, 12, 13, 14, 15, and 16. We see there are a total of seven
elements in set 𝑋. So this is the number of elements
that we’re choosing from.
We also know that 𝑎 is not equal
to 𝑏. So we need to find the number of
ways to choose two different elements from 𝑋 where order doesn’t matter. That is, we’re counting the number
of combinations. This means that 𝑛 of 𝑦, the
number of elements in set 𝑦, must be equal to seven choose two. Now, of course, the formula for 𝑛
choose 𝑟 is 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial. By letting 𝑛 be equal to seven and
𝑟 be equal to two, we can rewrite seven choose two as follows. It’s seven factorial over two
factorial times seven minus two factorial. The denominator of this expression
simplifies to two factorial times five factorial. We’re then going to rewrite seven
factorial as seven times six times five factorial. And two factorial is simply two
times one, so it’s two.
This then means we can divide by a
constant factor of five factorial, and we can also divide by two. So seven choose two simplifies to
seven times three divided by one, which is equal to 21. So, by establishing 𝑛 of 𝑦 as
being the number of ways of choosing two items from a total of seven where order
doesn’t matter, we have found that the number of elements in set 𝑦 is 21.
