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.
