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Question Number 115170 by bobhans last updated on 24/Sep/20

(1) G i v e n \frac{P_{n - 1}^{2 n + 1}}{P_{n}^{2 n - 1}} = \frac{3}{5}, f i n d n = ?

(2) i n h o w m a n y w a y s c a n 6 p e r s o n s

s t a n d i n a q u e u e ?

(3) h o w m a n y d i f f e r e n t 4 l e t t e r w o r d s

c a n b e f o r m e d b y u s i n g l e t t e r s o f

E D U C A T I O N u s i n g e a c h l e t t e r a t

m o s t o n c e ?

Answered by bemath last updated on 24/Sep/20

(1) \frac{\frac{(2 n + 1)!}{(n + 2)!}}{\frac{(2 n - 1)!}{(n - 1)!}} = \frac{3}{5} \to \frac{(2 n + 1)! (n - 1)!}{(2 n - 1)! (n + 2)!} = \frac{3}{5}

\frac{(2 n + 1) . 2 n . (n - 1)!}{(n + 2) (n + 1) n (n - 1)!} = \frac{3}{5}

\frac{4 n + 2}{(n + 2) (n + 1)} = \frac{3}{5}

20 n + 10 = 3 (n^{2} + 3 n + 2)

3 n^{2} - 11 n - 4 = 0

(3 n + 1) (n - 4) = 0 \to n = 4

Answered by bemath last updated on 24/Sep/20

(2)^{6} C_{6} \times 6! = 720

Answered by bemath last updated on 24/Sep/20

(3)^{9} C_{4} \times 4! = \frac{9!}{4! . 5!} \times 4! = \frac{9!}{5!}

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