8 men entered a lounge simultaneously. If each person shook hands with the other, then find the total no. of hand shakes?

Required number of ways = ²⁵C₃
= (25 x 24 x 23) / (1 x 2 x 3) = 2300

Required number of ways = ¹⁰C₃
= 10!/ {3! (10 - 3)!} = 10! / (3! 7!) = (10 x 9 x 8) / (3 x 2) = 120

Total no . of person = 8 + 7 = 15
No. of groups = ¹⁵C₆ = 15! / {6! (15 - 6)!} = 15! / (6! 9!)
= (15 x 14 x 13 x 12 x 11 x 10) / (6 x 5 x 4 x 3 x 2 x 1)
= 5005

? ⁵⁶P_{r + 6} : ⁵⁴P_{r + 3} = 30800 : 1
? 56! / (50 - r)! = (30800 x 54!) / (51-r!)
? 56 x 55 = 30800/(51 - r)
? 51 - r = 10
? r = 41

Total number of possible outcomes
= ¹²C₃ = 220
Number of events which do not contain blue marbles (3 marbles out of 7 marbles) = ⁷C₃ = 35
? Required probability = 1 - 35/220 = 37/44

ⁿ⁺²C₈ : ^n-2P₄ = 57 : 16
? {(n + 2 )! (n - 6)!} / {(n - 6)! (n - 2)! 8!} = 57/16
? (n + 2) (n + 1) n(n - 1) = 143640
? (n² + n - 2) (n² + n ) = 143640
? (n² + n )² - 2(n² + n) + 1 = 143641
? (n² + n - 1)² = (379)²
[? n² + n - 1 > 0]
? n² + n - 1 = 379
? n² + n - 380 = 0
? (n + 20) (n - 19) = 0
?n = 19 ( Since n is not negative)

⁵ P₂ = 5! / (5 - 2)! = 5 x 4 = 20

By formula, ⁿc_x = ⁿc_y
? x = y or x + y = n

Now, ⁵⁰c_r = ⁵⁰c_{r + 2}
? r + r + 2 = 50 or r = r + 2
? 2r = 48 [? r = r + 2 is not possible]
? r = 24

ⁿP₃ = 9240 ? n! / (n-3)! = 9240
? n(n - 1)(n - 2) = 9240
? n(n - 1)(n - 2) = 22 x 21 x 20
? n = 22

The word 'INHALE' has 6 distinct letters.
? Number of arrangements = n ! = 6 !
= 6 x 5 x 4 x 3 x 2 x 1
= 720