Question -
Answer -
Given:
P(15, r – 1) : P(16, r– 2) = 3 : 4
P(15, r – 1) / P(16, r– 2) = 3/4
By 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)!
So, from the question,
P(15, r – 1) / P(16, r– 2) = 3/4
Substituting theobtained 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
∴ r =14 [for, P(15, r – 1)]