Another way to count the nunibci ut nonricga-uve integral solmiorb to an equation of the form ¦«i *-«2 + •• ¦ * i» " m to reduce the pioblein lo
one of finding ike number of n-tupies (r,. v_;.....
y„) with 0 £ vi * *i X ¦ ¦ ¦ * v. S m. The reduction ic%uli> from letting y,^mXi+Xj + ¦•• + s,for
each I ¦ 1.2.....n. Use Mi approach to derive
a general formula for the number ot nunnegalive integral solution* to .«, + x, + -•• + *. — m.

To find another way to count the number of nonnegative integral solutions to an equation of the form x1+x2+.......+xn=m.

Given information:

The objective is to determine the number of nonnegative integral solutions to

x1+x2+.......+xn=m.

Concept used:

Select n integers for the n− tuples from the m integers (as 0≤y1≤y2≤........≤yn≤m).

The order of integers does not matter. So, use a combinations.

Calculation:

The number of nonnegative integral solutions is.

( n+m−1 n)=( n+m−1)!n!( n+m−1−n)!=( n+m−1)!n!( m−1)!=( n+m−1)⋅( n+m−2)⋅.....⋅( n+1)⋅n!n!( m−1)⋅( m−2)⋅....