Let S be the set of all seven-digit numbers that can be formed using the digits 0,1 and 2 . For example, 2210222 is in S, but 0210222 is NOT in S. Then the number of elements x in S such that at least one the digits 0 and 1 appears exactly twice in x, is equal to ______.
A. A
Let A " 0 " appear exactly twice. and B " 1 " appear exactly twice. A B " 0 " and " 1 " both appears exactly twice. n ( A )=- ------- = _ placing zero ^6 C _2(1)(2)^5= 6 5 2 2^5=480 for n ( B ) C-I : 1 at first place 1 ------- Number of ways = placing 1 ^6 C _1(1)(2)^5=192 C-II : 2 at first place 2 ------- Number of ways = placing 1 ^6 C _2(1)(2)^4 = 6 5 2 2^4=240 n(B)=240+192 for n ( A B ) ------- for n ( A B )= placing zero ^6 C _2(1) placing 1 ^5 C _2(1) 2at rest places 1 (1) = 6 5 2 5 4 2 =150 aligned & n ( A B )= n ( A )+ n ( B )- n ( A B ) \\ & =480+(192+240)-150 \\ & =762 aligned
Related: Mathematics — Permutation Combination · All PYQ Banks