# Question Video: Evaluating Combinations to Find an Unknown Set Size Mathematics

If ππΆββ + ππΆ_(π β 42) = 2ππΆββ, find π.

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### Video Transcript

If ππΆ42 plus ππΆπ minus 42 equals two times ππΆ43, find π.

Looking at this equation, we have three combinations, all of which have a set size of π, but a different size π. Recall that the definition for the number of combinations of size π taken from a set size π is given by π factorial over π factorial times π minus π factorial.

Now, one strategy for solving here would be to write each of these combinations in this expanded definition form. However, after we examine the two combinations on the left, we recall a property that we can use to simplify. And thatβs the symmetry property of combinations. It tells us that ππΆπ equals ππΆπ minus π. That is, the combination size π from set π will be equal to the combination of size π minus π of a size set π. By this property, both of the combinations on the left side of this equation are equal to each other.

This means we could rewrite our first term ππΆ42 as ππΆπ minus 42. After we bring everything else down, we can simplify to two times ππΆπ minus 42 equals two times ππΆ43. The twos on both sides cancel out. This implies that π minus 42 would be equal to 43. And if π minus 42 equals 43, then π would equal 85.

Because we know of the symmetry property, itβs also worth checking the second solution that could be possible. For ππΆπ equal to ππΆπ, the options are π equals π, which we found here, or π equals π minus π. That second solution would be π minus 42 equals π minus 43, which is not consistent. This means thereβs only one solution for π in this combination equation. If you wanted to check that this was correct, you could plug back in 85 for π and solve the combinations on your calculator.