If $^{21} C_{1} + 3.^{21} C_{3} + 5.^{21} C_{5} + . . . . + 19.^{21} C_{19} + 21.^{21} C_{21} = k$ then the number of prime factors of $k$ are

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Solution

$T_{r} = (2 r - 1) .^{21} C_{2 r - 1}$
$S = 11 \sum r = 1 T_{r} = 11 \sum r = 1 21 \cdot^{20} C_{2 r - 2} S = 21 (^{20} C_{0} +^{20} C_{2} + . . . . +^{20} C_{20}) S = 21 \times 2^{19} S = 3 \times 7 \times 2^{19}$
Total Prime factors $= 3$

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