Find the value of five choose zero
plus five choose one plus five choose two all the way up to five choose five.
To answer this question, we’re
going to recall two facts. 𝑛𝐶𝑟 or 𝑛 choose 𝑟 is found by
dividing 𝑛 factorial by 𝑟 factorial times 𝑛 minus 𝑟 factorial. But we also know that combinations
have symmetry such that 𝑛 choose 𝑟 is equal to 𝑛 choose 𝑛 minus 𝑟.
Now, let’s look at all of the terms
in our summation. We can see that 𝑛 here is going to
be equal to five. And so let’s begin by evaluating
five choose zero. In this case, 𝑟 is equal to
zero. So five choose zero is five
factorial over zero factorial times five minus zero factorial. Except, zero factorial is simply
one. So we get five factorial divided by
five factorial, which must also be equal to one. Due to this symmetry, we know that
this must also be equal to five choose five. And so we see that five choose zero
and five choose five are both equal to one.
We’re now going to work out five
choose one. That’s five factorial over one
factorial times five minus one factorial. One factorial is also one. So we get five factorial over four
factorial. But since five factorial is five
times four times three and so on, we can actually write it as five times four
factorial. This means we can divide through by
four factorial, and we find that five choose one is equal to five. And then using the symmetry, we
find that five choose four is also equal to five.
We’re now going to evaluate five
choose two and five choose three. For five choose two, we get five
factorial over two factorial times five minus two factorial, which gives us 10. Since there’s this symmetry, we
know this is also equal to five choose three. And so this means our sum is one
plus five plus 10 plus 10 plus five plus one, which is equal to 32.