A probability problem by Achal Jain
x 1 + x 2 + x 3 + 4 x 4 = 2 0
Find the number of non-negative integral solution to the equation above.
The answer is 536.
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1 solution
Nice Solution , but instead of adding 3 on the sides, you can easily substitute x 4 as 0,1,2,3,4 and 5. I did this cause 2 0 is a small number. I wanted to know about UK's Educational System, is there any way you can tell me about it?
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I added three to both sides so the sticks and stones method works more easily, as two sticks cannot then be in the same position because x 1 + 1 , x 2 + 1 and x 3 + 1 are positive.
x 1 + x 2 + x 3 + 4 x 4 = 2 0 so
( x 1 + 1 ) + ( x 2 + 1 ) + ( x 3 + 1 ) + 4 x 4 = 2 3
If x 4 = 0 , then we can imagine that we have 23 stones, and we have to put in two sticks to divide them up. (to find the values of x 1 , x 2 and x 3 )
The first stick has 22 possible positions, and the second has 21 possible positions, so there are 2 1 × 2 2 = 4 6 2 possible ways.
If x 4 = 1 , then there are 1 8 × 1 7 = 3 0 6 ways.
In total, there are k = 1 ∑ 6 ( 4 k − 2 ) ( 4 k − 3 ) = 1 0 7 2 ways.
However, we must divide by two to get rid of repeats, so the answer is 536.