Question

Find how many solutions there are to the given equation that satisfy the given condition. X1 + x2 + x3 = 22, each x; is a positive integer. Need Help? Read It Talk to a Tutor

Answer 1

210

Explanation:

Consider 22 counters ("stars") placed in a row. There are 21 spaces between them. The problem here amounts to finding the number of ways that 2 separators ("bars") can be put in those spaces, dividing the row into 3 parts.

The number of ways 21 objects can be chosen 2 at a time is ...

21C2 = 21!/(2!(21-2)!) = 21·20/(2·1) = 210

There are 210 positive integer solutions to the given equation.

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Here are the first few solutions, to give you the idea:

* | * | * * * ... ⇒ (x1, x2, x3) = (1, 1, 20)

* | * * | * * ... ⇒ (x1, x2, x3) = (1, 2, 19)

* | * * * | * ... ⇒ (x1, x2, x3) = (1, 3, 18)

Answer 2

Explanation:

Given that there are three variables satisfying the equation

`X1 + x2 + x3 = 22`

Here each x is given to be a positive integer

i.e. solution set for each of the variable can be any integer from 1 to 20 at most.(because if two other integers are 1 each third has to be 20)

Hence solution set can be of the form

`(x1,x2,x3) =(1,1,22) (1,2,21) (1,3,20).....`

`=(2,1,19) (2,2,18),...\\=(3,1,18) (3,2,17),....\\...\\...\\=(20,1,1)`

If x1 =1, there are 20 solution sets

If x1 =2,there are 19

...

If x1 =20 there is 1 set

Hence total solutions can be
`= 20+19+...+1\\=(20(21))/(2) =210`