If P(15, r – 1) : P(16, r – 2) = 3 : 4, find r.

1.	If P(15, r – 1) : P(16, r – 2) = 3 : 4, find r.
Answer» Given as P(15, r – 1) : P(16, r – 2) = 3 : 4 P(15, r – 1) / P(16, r – 2) = 3/4 On using the formula, P (n, r) = n!/(n – r)! P (15, r – 1) = 15! / (15 – r + 1)! = 15! / (16 – r)! P (16, r – 2) = 16!/(16 – r + 2)! = 16!/(18 – r)! Therefore, from the question, P(15, r – 1) / P(16, r – 2) = 3/4 On substituting the obtained values in above expression we get, [15! / (16 – r)!] / [16!/(18 – r)!] = 3/4 [15! / (16 – r)!] × [(18 – r)! / 16!] = 3/4 [15! / (16 – r)!] × [(18 – r) (18 – r – 1) (18 – r – 2)!]/(16 × 15!) = 3/4 1/(16 – r)! × [(18 – r) (17 – r) (16 – r)!]/16 = 3/4 (18 – r) (17 – r) = 3/4 × 16 (18 – r) (17 – r) = 12 306 – 18r – 17r + r² = 12 306 – 12 – 35r + r² = 0 r² – 35r + 294 = 0 r² – 21r – 14r + 294 = 0 r(r – 21) – 14(r – 21) = 0 (r – 14) (r – 21) = 0 r = 14 or 21 For, P(n, r): r ≤ n Hence r = 14 [for, P(15, r – 1)]

Answer»

Given as

P(15, r – 1) : P(16, r – 2) = 3 : 4

P(15, r – 1) / P(16, r – 2) = 3/4

On using the formula,

P (n, r) = n!/(n – r)!

P (15, r – 1) = 15! / (15 – r + 1)!

= 15! / (16 – r)!

P (16, r – 2) = 16!/(16 – r + 2)!

= 16!/(18 – r)!

Therefore, from the question,

P(15, r – 1) / P(16, r – 2) = 3/4

On substituting the obtained values in above expression we get,

[15! / (16 – r)!] / [16!/(18 – r)!] = 3/4

[15! / (16 – r)!] × [(18 – r)! / 16!] = 3/4

[15! / (16 – r)!] × [(18 – r) (18 – r – 1) (18 – r – 2)!]/(16 × 15!) = 3/4

1/(16 – r)! × [(18 – r) (17 – r) (16 – r)!]/16 = 3/4

(18 – r) (17 – r) = 3/4 × 16

(18 – r) (17 – r) = 12

306 – 18r – 17r + r² = 12

306 – 12 – 35r + r² = 0

r² – 35r + 294 = 0

r² – 21r – 14r + 294 = 0

r(r – 21) – 14(r – 21) = 0

(r – 14) (r – 21) = 0

r = 14 or 21

For, P(n, r): r ≤ n

Hence r = 14 [for, P(15, r – 1)]

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