Question Video: Using the Addition Rule of Counting Principles Mathematics

Video Transcript

What is the numerical expression that allows us to calculate in how many ways can a group of 10 people be formed from 10 boys and 12 girls such that the group has at least eight girls? Is it (A) 12 choose eight multiplied by 10 choose two plus 12 choose nine multiplied by 10 choose one plus 12 choose 10? Option (B) 12 choose eight multiplied by 10 choose two plus 12 choose nine multiplied by 10 choose one. Option (C) 12 choose eight multiplied by 10 choose one plus 12 choose nine multiplied by 10 choose zero. Option (D) 12 choose eight multiplied by 10 choose two multiplied by 12 choose nine multiplied by 10 choose one multiplied by 12 choose 10. Or option (E) 12 choose eight plus 10 choose two multiplied by 12 choose nine plus 10 choose one multiplied by 12 choose 10.

In this question, we are trying to select a group of 10 people from 10 boys and 12 girls, with a restriction that there must be at least eight girls. This means that we could select eight girls and two boys. Alternatively, we could select nine girls and one boy. Our final option would be to select 10 girls and no boys.

In this question, it doesn’t matter which order the boys and girls are selected. We recall that the number of ways of choosing 𝑟 items from 𝑛 items when order doesn’t matter is 𝑛 choose 𝑟. This means that choosing eight out of the 12 girls would be written 12 choose eight. As there are 10 boys in total, choosing two of these would be written 10 choose two. We can repeat this for nine girls and one boy and finally choosing 10 girls and zero boys.

As we want to select eight girls and two boys, we can use the fundamental counting principle. This states that the total number of outcomes is equal to the product of the number of outcomes of each event. Selecting eight girls and two boys is therefore equal to 12 choose eight multiplied by 10 choose two. Likewise, selecting nine girls and one boy is equal to 12 choose nine multiplied by 10 choose one. And finally, choosing 10 girls and zero boys is equal to 12 choose 10 multiplied by 10 choose zero. It is worth noting at this stage that 𝑛 choose zero and 𝑛 choose 𝑛 are both equal to one. This means that selecting 10 girls and zero boys is simply equal to 12 choose 10.

We require one of these three options to occur. And we know that the events are mutually exclusive. We recall that if 𝐴 and 𝐵 are mutually exclusive events, where 𝐴 has 𝑚 distinct outcomes and 𝐵 has 𝑛 distinct outcomes, the total number of outcomes is 𝑚 plus 𝑛. This is known as the addition rule. We need to add 12 choose eight multiplied by 10 choose two with 12 choose nine multiplied by 10 choose one with 12 choose 10. This corresponds to the answer given in option (A). The number of ways of selecting a group of 10 people from 10 boys and 12 girls such that the group has at least eight girls is 12 choose eight multiplied by 10 choose two plus 12 choose nine multiplied by 10 choose one plus 12 choose 10.