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

Answer:

  210

Step-by-step 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

Answer:

Step-by-step 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$= 20+19+...+1\\=\frac{20(21)}{2} =210$