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Dania Mueller

Dania Mueller

Answered question

2022-06-15

Let a , b , c be whole numbers so that
1 a 3
2 b 4
3 c 5
Find number of solutions of the equation, a + b + c = 10. Note: Please use FPC/PnC to answer this instead of the binomial theorem

Answer & Explanation

pyphekam

pyphekam

Beginner2022-06-16Added 27 answers

We need to find number of solutions to, A + B + C = 4 ,, where A , B , C are non-negative numbers and A 2 , B 2 , C 2. You can easily count case by case.
Alternatively, we could rewrite it as A = 2 x , B = 2 y , C = 2 z and find number of solutions to
x + y + z = 2 , x 2 , y 2 , z 2
Using stars and bars, we get ( 4 2 ) = 6 solutions.
Now coming back to counting number of solutions to A + B + C = 4, please see the cases below. Can you count number of solutions in each case?
- If A = 0, we have B + C = 4
- If A = 1, we have B + C = 3
- If A = 2, we have B + C = 2

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