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

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Correct answer: 0

Let A arrow " 0 " appear exactly twice. and B arrow " 1 " appear exactly twice. A B arrow " 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

Let S be the set of all seven-digit numbers that can be formed...

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