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The value of the expression ∑ k = 0 27 k ⋅ C k   27 1 3 k 2 3 27 - k is equal to

Correct answer

9

Step-by-step solution

Given expression can be rewritten as ∑ k = 1 27 k ⋅ 27 k C k - 1   26 1 3 1 3 k - 1 2 3 26 - ( k - 1 ) = 9 ∑ k = 1 27 C k - 1   26 1 3 k - 1 2 3 26 - ( k - 1 ) = 9 ∑ k - 1 = 0 26 C k - 1   26 2 3 26 - ( k - 1 ) 1 3 k - 1 = 9 2 3 + 1 3 26 = 9 × 1 = 9

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