Odd solutions to a sum

MediumCombinatorics~5m

How many solutions in non-negative odd integers $(x_1, \dots, x_6)$ satisfy $x_1 + x_2 + \dots + x_6 = 96$ ? The count equals $\dbinom{n}{k}$ with $k < \tfrac{n}{2}$ ; report $n\cdot k$ .

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Odd and non-negative means $x_i = 2y_i + 1$ with $y_i \ge 0$ . Substitute into the sum.
You get $\sum y_i = 45$ over $6$ variables - a stars-and-bars count $\binom{50}{5}$ .

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Odd solutions to a sum

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