Question Video: Using Combinations to Solve a Problem Expressed Using Set Notation | Nagwa Question Video: Using Combinations to Solve a Problem Expressed Using Set Notation | Nagwa

Question Video: Using Combinations to Solve a Problem Expressed Using Set Notation Mathematics • Third Year of Secondary School

Let 𝑋 = {𝑥 : 𝑥 ∈ ℤ, 10 ≤ 𝑥 ≤ 16} and 𝑦 = {{𝑎, 𝑏} : 𝑎, 𝑏 ∈ 𝑋, 𝑎 ≠ 𝑏}. Determine the value of 𝑛(𝑦), where 𝑛(𝑦) is the number of elements in 𝑦.

02:30

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.

Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

  • Interactive Sessions
  • Chat & Messaging
  • Realistic Exam Questions

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy