If ^{2n + 1}P _{n - 1} : ^{2n - 1}P _n = 3 : 5, then n is equal to?

³⁵C_{n + 7} = ³⁵C_{4n - 2}
We know that if ⁿC_x = ⁿC_y then, Either x = y or x + y = n

Case 1: x + y = n
? n + 7 + 4n - 2 = 35
? 5n + 5 = 35
? 5n = 30
? n = 6

Case 2: x = y
? n + 7 = 4n - 2
? 4n - n = 2 + 7
? 3n = 9
? n = 3

ⁿC_{r - 1} = 36, ⁿC_r = 84 and ⁿC_{r + 1} = 126
? ⁿC_r / ⁿC_{r - 1} = 84/36
? 3n - 10r + 3 = 0 ....(i)

ⁿC_{r + 1} / ⁿC_r = 126/84
? 2n - 5r - 3 = 0 .............(ii)

From (i) and (ii) r = 3, n = 9

We have 9 letters 3 a's, 2 b's and 4 c's. These 9 letters can be arranged in 9! / (3! 2! 4!) = 1260 ways.

Required number of ways = (2 + 1) (3 + 1) (4 + 1) - 1 = 59

12 persons can be seated around a round table in 11! ways. The total number of ways in which 2 particular persons sit side by side = 10! x 2!.

Hence, the required number of arrangements = 11! - 10! x 2! = 9 x (10!)

We know that, ⁿP_r = ⁿC_rr!
? ⁿP_r = 720ⁿC_r
? ⁿC_r.r! = 720ⁿC_r
? r! = 720
? r = 6

The hundred place will be reserved for 3 or 2
5 digits are free to fill rest two places i,e., of tens and unit.

Number of required 3 digit number = 2 x ⁵C₂ = 20

The number of ways of selecting 3 points out of 12 points is ¹²C₃. The number of ways of selecting 3 points out of 7 points, on the same straight line is ⁷C₃, Hence, the number of triangle formed will be ¹²C₃ - ⁷C₃ = 210 - 35 = 185.

Total number of letters = 5
Number of vowels = 2.
If we consider both vowel as a one letter then,
Required number = 4! 2! = 48.

We may choose 1 officer and 5 jawans or 2 officers and 4 jawans ......... or 4 officers and 2 jawans.

So Required answer = [⁴C₁ x ⁸C₅] + [⁴C₂ x ⁸C₄] + [⁴C₃ x ⁸C₃] + [⁴C₄ x ⁸C₂]
= 224 + 420 + 224 + 28 = 896.