Question Video: Using the Addition Rule of Counting Principles Mathematics • Third Year of Secondary School

Video Transcript

Which of the following expressions shows how to calculate the number of ways that a group of six people can be formed from five teachers and 10 parents such that the group has at least one parent and at least one but fewer than four teachers? Is it option (A) five choose four multiplied by 10 choose two plus five choose three multiplied by 10 choose three plus five choose two multiplied by 10 choose four plus five choose one multiplied by 10 choose five? Option (B) five choose four plus 10 choose two plus five choose three plus 10 choose three plus five choose two plus 10 choose four plus five choose one plus 10 choose five. Option (C) five choose three multiplied by 10 choose three plus five choose two multiplied by 10 choose four plus five choose one multiplied by 10 choose five. Option (D) five choose three plus 10 choose three plus five choose two plus 10 choose four plus five choose one plus 10 choose five. Or option (E) five choose three times 10 choose three times five choose two times 10 choose four times five choose one times 10 choose five.

In this question, we are trying to select six people from a group of five teachers and 10 parents. This is subject to two restrictions. Firstly, the group must have at least one parent and, secondly, at least one but fewer than four teachers. As the group must have fewer than four teachers, we could have three teachers and three parents. Alternatively, we could have two teachers and four parents. Finally, since there must be at least one teacher, we can have one teacher and five parents.

We recall that the number of ways of choosing 𝑟 items from 𝑛 items when order doesn’t matter can be written 𝑛 choose 𝑟. And in this question, it doesn’t matter which order we select the teachers and parents. This means that choosing three out of the five teachers would be written five choose three. And choosing three out of the 10 parents would be written 10 choose three. The fundamental counting principle tells us that the number of ways of choosing three teachers and three parents is found by multiplying these values. This gives us five choose three multiplied by 10 choose three. And this is the number of ways that we can choose three teachers and three parents.

Repeating this for two teachers and four parents, we have five choose two multiplied by 10 choose four. And finally, the number of ways of choosing one teacher and five parents is five choose one multiplied by 10 choose five. We require one of these three options to occur, and we know that the events are mutually exclusive. Recalling that if 𝐴 and 𝐵 are mutually exclusive events, where 𝐴 has 𝑚 distinct outcomes and 𝐵 has 𝑛, the total number of outcomes is 𝑚 plus 𝑛. We can add our three values: five choose three multiplied by 10 choose three plus five choose two multiplied by 10 choose four plus five choose one multiplied by 10 choose five.

This corresponds to option (C). The number of ways of selecting a group of six people from five teachers and 10 parents such that the group has at least one parent and at least one but fewer than four teachers is five choose three multiplied by 10 choose three plus five choose two multiplied by 10 choose four plus five choose one multiplied by 10 choose five.