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

Find the possible values of π which satisfy the equation 21C_(π) = 21Cββ.

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### Video Transcript

Find the possible values of π which satisfy the equation 21Cπ equals 21C15.

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

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