The number of ways of division or distribution of n identical things into r different  is n+r-1Cr-1    .Prove it.

no. of ways of dividing n identical things to r different people(say)

then divisions will vary only with the quantity given to the persons (since things are identical)

so let x1 , x2 ,x3, .......... xR be the quantity given to person 1 person 2,person 3...... & so on till person r resp.

so we have to find the no. of non negative integral solns of  x1 + x2 + x3 + ....... + xR = n

which is coeff . of x^n in (1+x+x^2....... + x^n)^r

= coeff. of x^n in[ (1-x^(n+1))/(1-x)]^r

= coeff .of x^n in (1-x)^-r       

coeff of x^n in (1 + rC1x +  r+1C2x^2  +........... + n+r-1Cnx^n )

= n+r-1Cn = n+r-1Cr-1