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Question Video: Using the Properties of Combinations to Find the Value of an Unknown Mathematics

Find the possible values of 𝑟 which satisfy the equation 21C_(𝑟) = 21C₁₅.

01:41

Video Transcript

Find the possible values of 𝑟 which satisfy the equation 21C𝑟 equals 21C15.

Of course, we could take the definition for combinations, which tell us 𝑛C𝑟 equals 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial. And then we could expand each of these combinations to try and solve. However, sometimes we can solve problems in a simpler and more straightforward manner by being familiar with properties of combinations.

One such property is related to the symmetry of combinations. This tells us that 𝑛C𝑟 equals 𝑛C𝑛 minus 𝑟. If we apply that here, for our first term we’ll let 𝑛 be equal to 21 and 𝑟 be equal to 𝑟. And again for the right-hand side, 𝑛 equals 21, but 15 will be equal to 𝑛 minus 𝑟 to set up the symmetry of combinations. If 15 equals 𝑛 minus 𝑟 and 𝑛 equals 21, we subtract 21 from both sides and see that negative six equals negative 𝑟 and that 𝑟 equals six.

However, we should notice that there is a possible solution as well where 𝑟 equals 15, because of course 21C15 equals 21C15, meaning we have a possible solution at 𝑟 equals 15 and a possible solution at 𝑟 equals six.

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