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Question
Find r if `""^11"C"_4 + ""^11"C"_5 + ""^12"C"_6 + ""^13"C"_7 = ""^14"C"_"r"`
Solution
Use `""^"n""C"_"r" + ""^"n""C"_("r" - 1) = ""^("n" + 1)"C"_"r"`
∴ `""^11"C"_4 + ""^11"C"_5 + ""^12"C"_6 + ""^13"C"_7 = ""^14"C"_"r"`
∴ `(""^12"C"_5 + ""^12"C"_6) + ""^13"C"_7 = ""^14"C"_"r"`
∴ `(""^13"C"_6 + ""^13"C"_7) = ""^14"C"_"r"`
∴ `""^14"C"_7 = ""^14"C"_"r"`
If nCx = nCy, then either x = y or x = n − y
∴ r = 7 or r = 14 − 7 = 7
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