Question

The number of non-negative integral solutions of $x_1+x_2+x_3+x_4=10$ where not all the $x_i$ are different is

• A. $286$
• B. $282$
• C. $166$
• D. $168$

Answer 1

The correct option is C $166$

Number of solutions without any conditions = ${}^{10+4-1}{C}_{4-1}=286$
Now, all terms are different can be attained in the following ways:
(a) $0,1,4,5:4!=24$ ways
(a) $0,1,3,6:4!=24$ ways
(a) $0,1,2,7:4!=24$ ways
(a) $0,2,3,5:4!=24$ ways
(a) $1,2,3,4:4!=24$ ways
Hence, total $=24\times 5=120$ ways.
Hence, answer $=286-120=166$
Hence, (C) is correct.