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

Find the possible values of π‘Ÿ which satisfy the equation 21C_(π‘Ÿ) = 21C₁₅.


Video Transcript

Find the possible values of π‘Ÿ which satisfy the equation 21Cπ‘Ÿ equals 21C15.

Of course, we could take the definition for combinations, which tell us 𝑛Cπ‘Ÿ equals 𝑛 factorial over π‘Ÿ factorial times 𝑛 minus π‘Ÿ 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 𝑛Cπ‘Ÿ equals 𝑛C𝑛 minus π‘Ÿ. If we apply that here, for our first term we’ll let 𝑛 be equal to 21 and π‘Ÿ be equal to π‘Ÿ. And again for the right-hand side, 𝑛 equals 21, but 15 will be equal to 𝑛 minus π‘Ÿ to set up the symmetry of combinations. If 15 equals 𝑛 minus π‘Ÿ and 𝑛 equals 21, we subtract 21 from both sides and see that negative six equals negative π‘Ÿ and that π‘Ÿ equals six.

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

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