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Question
find the value of `sum_("r" = 1)^4 ""^(21 - "r")"C"_4 + ""^17"C"_5`
Solution
`sum_("r" = 1)^4 ""^(21 - "r")"C"_4 + ""^17"C"_5`
= `""^20"C"_4 + ""^19"C"_4 + ""^18"C"_4 + ""^17"C"_4 + ""^17"C"_5`
= `""^20"C"_4 + ""^19"C"_4 + ""^18"C"_4 + ""^18"C"_5-""^17"C"_5+""^17"C"_5` ........[nCr + nCr−1 = n+1Cr]
= `""^20"C"_4 + ""^19"C"_4 + ""^18"C"_4+""^18"C"_5`
= `""^20"C"_4 + ""^19"C"_4 + ""^19"C"_5`
= `""^20"C"_4 + ""^20"C"_5`
= `""^21"C"_5`
=`sum_("r" = 1)^4 ""^(21 - "r")"C"_4 + ""^17"C"_5=""^21"C"_5`
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