Question

For the equation $x+y+z+w=19,$ the number of positive integral solutions is equal to:

• A. The number of ways in which 15 identical things can be distributed among 4 persons
• B. the number of ways in which 19 identical things can be distributed among 4 persons
• C. coefficient of $x^{19}$ in ${(x^0+x^1+x^2+...+x^{19})}^4$
• D. coefficient of $x^{19}$ in ${(x+x^2+x^3+...+x^{19})}^4$

Answer 1

Answer: (A), (D)

The number of ways in which 15 identical things can be distributed among 4 persons
coefficient of $x^{19}$ in ${(x+x^2+x^3+...+x^{19})}^4$

Number of ways in which 19 identical items can be divided in 4 groups(x,y,z,w) so that no group contains less than 1 items and more than 19
It is given by the coefficient of
$x^{19}$ in $(x^1+x^2+...x^{19}{)}^4$
$={}^{18}{C}_3$
$={}^{15+4-1}{C}_{4-1}$
= The number of ways in which 15 identical things can be distributed among 4 persons.