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Sum

Find the value of `"r" (""^56"C"_("r" + 6)): ""^54"P"_("r" - 1)`= 30800 : 1

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#### Solution

`""^56"P"_("r" + 6): ""^54"P"_("r" + 3)` = 30800 : 1

∴ `(""^56"P"_("r" + 6))/(""^54"P"_("r" + 3)) = 30800/1`

∴ `((56!)/((56 - "r" - 6)!))/((54!)/((54 - "r" - 3)!)` = 30800

∴ `(56!)/((54 - "r" - 6)!) xx ((54 - "r" - 3)!)/(54!)` = 30800

∴ `(56!)/((50 - "r")!) xx ((51 - "r")!)/(54!)` = 30800

∴ `(56 xx 55 xx 54!)/((50 - "r")!) xx ((51 - "r")(50 - "r")!)/(54!)` = 30800

∴ 51 – r = `30800/(56 xx 55)`

∴ 51 – r = 10

∴ r = 51 – 10

∴ r = 41

Concept: Introduction of Permutations and Combinations

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