Lesson 8 of 11 · 10 min
Properties of combinations
NCERT §6.4
Picking 8 of the 10 relay reserves to travel is the same as picking the 2 who stay behind. That simple idea saves a lot of arithmetic.
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nCr = nC(n − r): choosing r objects to take is the same as choosing n − r objects to leave. So 10C8 = 10C2 = 45.
If nCa = nCb, then a = b or a + b = n. If nC7 = nC5, then n = 12, and 12C2 = 66.
Pascal's rule: nCr + nC(r − 1) = (n + 1)Cr. For example 5C2 + 5C1 = 10 + 5 = 15 = 6C2.
The values nC0, nC1, …, nCn rise to the middle and fall back symmetrically. For n = 8 they are 1, 8, 28, 56, 70, 56, 28, 8, 1, with the largest, 70, at r = 4.
Taking at least one of 6 different things: 6C1 + 6C2 + … + 6C6 = 6 + 15 + 20 + 15 + 6 + 1 = 63.