Finding Number Of Solutions Of Equations - Practice Questions & MCQ

To find the  solutions of a + b + c = 6 such that a,b,c are non-negative integers

Since zero is included, we have a ≥ 0, b ≥ 0, c ≥ 0

Number of solutions of a + b + c = 6 is equivalent to the number of ways of distributing 6 identical things into 3 different people, which can be achieved by arranging 6 objects and 3 - 1 = 2 partitions in a row. The number of objects before the first partition is given to first person, the number of objects in between the two partitions is given to second person and number of objects after second partition is given to third person.

Number of ways of doing this arrangement is  ,  which can also be written as 

Hence the total number of solutions

Generalized formula

For whole number solutions of , we have the formula .

Generalized formula

The natural number solutions of  are 

Q: Find the Natural number of solutions of a + b + c = 6 such that a,b,c are natural numbers

Solution:

Since zero is included, we have a ≥ 1, b ≥ 1, c ≥ 1

Let us define new variables

a' + 1 = a

b' + 1 = b

c' + 1 = c

Now if a', b', c' are whole numbers then a, b, c will be natural numbers.

So the equation becomes a' + 1 + b' + 1 + c' + 1 = 6

a' + b' + c' = 3

So whole number solutions of this equation equal natural number solutions of a + b + c = 6 

Hence the total number of solutions