MathDB

Let

\ell = 1

M = 23

N = 45

, and

u = 67

. Compute the number of ordered pairs of nonnegative integers

(X, Y)

with

X \leq M - \ell

and

Y \leq N + u

such that the sum

\sum_{k=\ell}^{u} \binom{X + k}{M}\cdot\binom{Y - k}{N}

is divisible by

89

(for integers

a

and

b

, define the binomial coefficient

\tbinom{a}{b}

to be the number of

b

-element subsets of any given

a

-element set, which is

0

when

a < 0

b < 0

, or

b > a

Problem Statement