Question Video: Solving Using Combination Mathematics

Video Transcript

Evaluate 23 𝐶 19.

In order to answer this question, we begin by recalling the formula for combinations. 𝑛𝐶𝑘 is equal to 𝑛 factorial divided by 𝑛 minus 𝑘 factorial multiplied by 𝑘 factorial. In this question, 𝑛 is equal to 23 and 𝑘 is equal to 19. 23 𝐶 19 is therefore equal to 23 factorial divided by 23 minus 19 factorial multiplied by 19 factorial. The denominator simplifies to four factorial multiplied by 19 factorial. We can rewrite the numerator as 23 multiplied by 22 multiplied by 21 multiplied by 20 multiplied by 19 factorial. And by dividing the numerator and denominator by 19 factorial, we have 23 multiplied by 22 multiplied by 21 multiplied by 20 divided by four factorial.

Writing four factorial as four multiplied by three multiplied by two multiplied by one, we can reduce the factors as shown. This leaves us with 23 multiplied by 11 multiplied by seven multiplied by five, which is 8,855. 23 𝐶 19 is equal to 8,855.