1.

The number of `n` digit number formed by using digits `{1,2,3}` such that if `1` appears, it appears even number of times, isA. `2^(n)+1`B. `(1)/(2)(3^(n)+1)`C. `(1)/(2)(3^(n)-1)`D. `(1)/(2)(2^(n)-1)`

Answer» Correct Answer - B
`(b)` Required number of numbers `="(n)C_(0)(2)^(n)+"(n)C_(2)(2)^(n-2)+…`
We know that
`(1+x)^(n)+(x-1)^(n)=2sum_(k=0)^(n)"(n)C_(k)x^(k)`
Hence, put `x=2`, get the result
`2["^(n)C_(0)(2)^(n)+^(n)C_(2)(2)^(n-2)+^(n)C_(2)(2)^(n-4)+^(n)C_(6)(2)^(n-6)+...]`


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