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.