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Question
Prove that `""^35"C"_5 + sum_("r" = 0)^4 ""^((39 - "r"))"C"_4` = 40C5
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Solution
`""^35"C"_5 + sum_("r" = 0)^4 ""^((39 - "r"))"C"_4 = ""^35"C"_5 + ""^((39 - 0))"C"_4 + ""^((39 - 1))"C"_4 + ""^((39 - 2))"C"_4 + ""^((39 - 3))"C"_4 + ""^((39 - 4))"C"_4`
= `""^35"C"_4 + ""^39"C"_4 + ""^38"C"_4 + ""^37"C"_4 + ""^36"C"_4 + ""^35"C"_4` ......(1)
`""^"n""C"_("r" - 1) + ""^"n""C"_"r" = ""^(("n" + 1))"C"_"r"`
(1) ⇒ `""^35"C"_5 + sum_("r" = 0)^4 ""^((39 - "r"))"C"_4`
= (35C5 + 35C4) + 39C4 + 38C4 + 38C4 + 37C4 + 36C4
= `""^((35 + 1))"C"_5 + ""^36"C"_4 + ""^39"C"_4 + ""^38"C"_4 + ""^37"C"_4`
= (36C5 + 36C4) + 39C4 + 38C4 + 37C4
= `""^((36 + 1))"C"_5 + ""^37"C"_4 + ""^39"C"_4 + ""^38"C"_4`
= (37C5 + 37C4) + 39C4 + 38C4
= `""^((37 + 1))"C"_5 + ""^38"C"_4 + ""^39"C"_4`
= (38C5 + 38C4) + 39C4
= 39C4 + 39C4
= `""^((39 + 1))"C"_5`
= 40C5
= R.H.S
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