Question Video: Using the Properties of Combinations to Evaluate Combinations Mathematics

Video Transcript

Find the value of four choose zero minus four choose one plus four choose two minus four choose three plus four choose four.

Remember, we know that combinations have symmetry such that 𝑛𝐶𝑟 is equal to 𝑛𝐶𝑛 minus 𝑟. And so this means that four 𝐶 zero will be equal to four 𝐶 four and four 𝐶 one will be equal to four 𝐶 three. In fact, we can go further than this and say that 𝑛𝐶 zero and 𝑛𝐶𝑛 are both equal to one. So we find four 𝐶 zero and four 𝐶 four are equal to one. Since 𝑛𝐶𝑟 is 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial, we find that four choose one is four factorial over one factorial times three factorial, which is equal to four, giving us four choose one is four and four choose three is four.

Let’s finally evaluate four choose two. It’s four factorial over two factorial times two factorial, which is equal to six. And so we’re ready to evaluate this alternating sum. We could use summation notation as shown. And we get one minus four plus six minus four plus one, which is equal to zero. And so the value of our alternating sum is zero. In fact, once again, we have a general rule. And that is the alternating sums of 𝑛 choose 𝑟 are equal to zero. In general, we can say that the sum from 𝑟 equals zero to 𝑛 of negative one to the power of 𝑟 times 𝑛 choose 𝑟 is equal to zero.