Let S = {1, 2, 3, ……., 9}. For k = 1, 2, …, 5, let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N1 + N2 + N3 + N4 + N5 =

Engineering

Mathematics

Permutation and Combination Basic Definitions

Question

Let S = {1, 2, 3, ……., 9}. For k = 1, 2, …, 5, let N_k be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N₁ + N₂ + N₃ + N₄ + N₅ =

126

125

210

252

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$\begin{matrix} 5 odd & + & 4 even \\ 1 & 4 \\ 2 & 3 \\ 3 & 2 \\ 4 & 1 \\ 5 & 0 \end{matrix}$

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Same as taking any 5 out of 9 = ⁹C₅ = ⁹C₄ = 126. Ans.