Question
Sat July 21, 2012 By:

# Find the no. of non-negative integral solutions of x1 + x2 + x3 + 4x4=20. Ans should be 536. Please answer by Sunday

Sun July 22, 2012

To solve this question, you have to consider some cases.

case I : let x4=0

x1+x2+x3=20.

You have to distribute 20 identical objects into 3 groups. No. of solutions to this = (20+3-1)C(3-1)=22C2

case II : let x4=1

x1+x2+x3=16

You have to distribute 16 identical objects into 3 groups. No. of solutions to this = (16+3-1)C(3-1)=18C2

case III : let x4=2

x1+x2+x3=12

You have to distribute 12 identical objects into 3 groups. No. of solutions to this = (12+3-1)C(3-1)=14C2

case IV : let x4=3

x1+x2+x3=8

You have to distribute 8 identical objects into 3 groups. No. of solutions to this = (8+3-1)C(3-1)=10C2

case V : let x4=4

x1+x2+x3=4

You have to distribute 4 identical objects into 3 groups. No. of solutions to this = (4+3-1)C(3-1)=6C2

case VI : let x4=5

x1+x2+x3=0

You have to distribute 0 identical objects into 3 groups. No. of solutions to this = 2C2 = 1

Hence total number of solutions = 1 +6C2 + 10C2 + 14C2 + 18C2 + 22C2

= 536

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