If P ( n , r ) = 1680 and C ( n , r ) = 70 , then 69 n + r ! =

Question

If $P (n, r) = 1680$ and $C (n, r) = 70$ , then $69 n + r! =$

Smart Achievers · Accepted Answer

(2)

$P (n, r) = 1680 and C (n, r) = 70$

$C (n, r) = \frac{n!}{r! (n - r)!} = 70$

$\Rightarrow \frac{P (n, r)}{r!} = 70 \Rightarrow \frac{1680}{r!} = 70 \Rightarrow r! = 24 = 4!$

$∴$ $r = 4$

$Now, P (n, r) = \frac{n!}{(n - r)!} = 1680$

$\frac{n!}{(n - 4)!} = 1680 \Rightarrow n (n - 1) (n - 2) (n - 3) = 1680$

$\Rightarrow$ $n (n - 1) (n - 2) (n - 3) = 8 \times 7 \times 6 \times 5$

$After comparing both sides, n = 8$

$Now, 69 n + r! = (69 \times 8) + 4! = 552 + 24 = 576$

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