Question Video: Evaluating the Quotient of Combinations Mathematics

Video Transcript

Evaluate seven 𝐶 two divided by eight 𝐶 six.

In this problem, we are given the quotient of two combinations. We recall that the combination 𝑛𝐶𝑘 represents the number of different ways to select 𝑘 objects out of a total of 𝑛 distinct objects. When dealing with combinations, the order does not matter.

To evaluate this expression, we begin by recalling the formula for combinations. 𝑛𝐶𝑘 equals 𝑛 factorial divided by 𝑛 minus 𝑘 factorial multiplied by 𝑘 factorial. To answer this question, we will evaluate each combination separately then divide their values.

Let’s begin with seven 𝐶 two. In this case, 𝑛 equals seven and 𝑘 equals two. We have to be careful not to mix up the 𝑛-value and the 𝑘-value during our computations. Seven 𝐶 two is therefore equal to seven factorial divided by seven minus two factorial multiplied by two factorial. The denominator simplifies to five factorial multiplied by two factorial. We can rewrite the numerator as seven multiplied by six multiplied by five factorial. Then, by dividing the numerator and denominator by five factorial, we have seven multiplied by six divided by two factorial. Next, we write two factorial as two multiplied by one. This leaves us with 42 divided by two, which is 21.

Now we will evaluate the combination in the denominator of the given expression, which has 𝑛 equal to eight and 𝑘 equal to six. Eight 𝐶 six therefore equals eight factorial divided by eight minus six factorial multiplied by six factorial. The denominator simplifies to two factorial multiplied by six factorial. Then, we can write eight factorial as eight multiplied by seven multiplied by six factorial. This allows us to divide both the numerator and the denominator by six factorial, leaving us with eight multiplied by seven divided by two factorial. We reach a final value of 28 by taking the product of eight and seven divided by the product of two and one.

To evaluate the original quotient, all that is left is to simplify the fraction 21 over 28. We reach our final answer of three-fourths by dividing both the numerator and the denominator by seven. We have shown that three-fourths is the value of seven 𝐶 two divided by eight 𝐶 six.