In how many ways, the letters of the word 'BANKING' can be arranged?

Required number of arrangements = 6 ! / 3 ! [? S has come thrice ]
= [ 6 x 5 x 4 x 3! ] /3 !
= 120

Number of arrangements n ! /( p ! q ! r ! )
Total letters = 6, but R has come twice.
So, required number of arrangements
= 6 ! / 2 ! = (6 x 5 x 4 x 3 x 2 !) / 2 ! = 360

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

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

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

Total number of letters = 7, but N has come twice.
So, required number of arrangements = 7 ! / 2 !
= [7 x 6 x 5 x 4 x 3 x 2! ] / 2!
= 2520

The required different ways = 7 ! / 2 !
= 7 x 6 x 5 x 4 x 3 x 2!/2!
= 2520

Total number of handshakes = 12 x 12 = 144

Required number of lines = ⁿC₂
= ¹⁵C₂
= (15 x 14)/2
= 105

Total number of books = a + 2b + 3c + d . Since there are 'b' copies of each of two books, 'c' copies of each of three books and single copy of 'd' book.
Therefore, the total number of arrangements is = (a + 2b + 3c + d )! / {a! (b!)² (c!)³}