Question

Prove that `""^35"C"_5 + sum_("r" = 0)^4 ""^((39 - "r"))"C"_4` = ⁴⁰C₅

Answer 1

`""^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