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Question
Prove that 15C3 + 2 × 15C4 + 15C5 = 17C5
Solution
To prove 15C3 + 2 × 15C4 + 15C5 = 17C5
15C3 + 2 × 15C4 + 15C5 = (15C3 + 15C4) + (15C4 + 15C5)
`""^"n""C"_("r" - 1) + ""^"n""C"_"r" = ""^(("n" + 1))"C"_"r"`
15C3 + 2 × 15C4 + 15C5 = `""^((15 + 1))"C"_4 + ""^((15 + 1))"C"_5`
= 16C4 + 16C5
= `""^((16 + 1))"C"_5`
= 17C5
∴ 15C3 + 2 × 15C4 + 15C5 = 17C5
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