Let S =\ 1,2,3,4,5,6,7,8,9\ . Let x be the number of 9 -digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let y be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,
D. 21 x=4 y
x counts 9-digit numbers using \ 1,2,3,4,5,6,7,8,9\ with exactly one digit repeated twice (8 distinct digits, one twice, one absent). Choose digit appearing twice: 9 ways. Choose absent digit: 8 ways. Arrange: 9! 2! ways. x = 9 8 9! 2! = 72 181440 = 13043520. y counts 9-digit numbers with exactly two digits each repeated twice (7 distinct digits, two appear twice, two absent). Choose 2 digits for repetition: 9 2 ways. Choose 2 absent digits: 7 2 ways. Arrange: 9! 2! 2! ways. y = 36 21 9! 4 = 756 9! 4 . x y = 72 9!/2 756 9!/4 = 72 4 756 2 = 288 1512 = 4 21 . Therefore 21x = 4y
Related: Mathematics — Permutation Combination · All PYQ Banks