The number of ways of selecting $n$ objects out of $3 n$ objects, $n$ of which are alike and the rest different is

2^{2 n} - \sum 0 \leq i < j \leq n \sum C_{i} C_{j}

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2^{2 n - 1} - \sum 0 \leq i < j \leq n \sum C_{i} C_{j}

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\sum 0 \leq i < j \leq n \sum C_{i} C_{j}

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None of these

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Solution

The correct option is D $2^{2 n} - \sum 0 \leq i < j \leq n \sum C_{i} C_{j}$
Total number of objects $= 3 n$
Total number of 'different' objects $= 3 n - n = 2 n$
Hence number of ways of selecting a object from 2n objects is $2^{2 n}$
Hence the required permutation of selecting n objects from 3n objects with n alike objects is
$2^{2 n} - \sum \sum_{0 \leq i \leq j \leq n} C_{i} . C_{j}$ .

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