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 $n\cdot k$ .

Show hints (2)+

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 $\binom{50}{5}$ .

Answer

Reveal answer →

250

Want the full step-by-step worked solution? It's part of Premium - along with a worked solution for every question in the bank.

Practise free Unlock solutions

Asked at: Jane Street, Citadel

Odd solutions to a sum

Answer

Related questions