InterviewSolution
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InterviewSolution
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There are p letters a, q letters b, r letters C.The number of ways of selecting k letters out of these if `p lt k lt q lt r` isA. `3^(n-1)`B. `n 3^(n)`C. `n3 ^(n-1)`D. `2 n ^(n-1)` |
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Answer» Correct Answer - c If P `cup` Q conains exaclty one element , both P and Q must be non-enpty. Thus, if P has r elements, Q must have exactly one of these r elements and any number of elements from among the reamaining (n-r ) elements in A, so that the number of ways of choosing Q is `""^(r)C_(1) xx2^(n-r)` . But, P can be chosen in `""^(n)C_(r)` ways. Therefore, P and Q can be chosen in `""^(n)C_(1) xx""^(r)C_(1)xx2^(n-r)` , when P contains r elements. As r can very from 1 to n . Therefore, P and Q in general can be chosen in `sum_(r=1)^(n) ""^(n)C_(r) xx""^(n)C_(r)xx2^(n-r)` `=sum_(r=1)^(n) ""^(n)C_(r)r2^(n-r)` `=sum_(r=1)^(n) (n)/(r) ""^(n-r)C_(r-1).r.2^(n-r)` "" `[because ""^(n)C_(r)= (n)/(r) ""^(n-1)C_(r - 1)]` `nsum_(r=0)^(n) ""^(n-1)C_(r-1)2^(n-r)` `nsum_(r=0)^(n) ""^(n-1)C_(r-1)2^((n-r)-(r-0)) = n(1 + 2)^(n+1) = n3^(n-1)` |
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