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Find the number of non-negative integral solutions to the system of equations x + y + z + u + t = 20 and x + y + z = 5.
- a)220
- b)336
- c)44
- d)528
Correct answer is option 'B'. Can you explain this answer?
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Find the number of non-negative integral solutions to the system of eq...
n identical things can be distributed among r persons in n+x-1Cx-1
Most Upvoted Answer
Find the number of non-negative integral solutions to the system of eq...
Taking x+y+z=5
There can be 21 cases satisfying this equation.
0+0+5=5 (0,0,5 can be assigned to x, y, z in 3 ways)
0+1+4=5 (0,1,4 can be assigned to x, y, z in 6 ways)
0+2+3=5 (0,2,5 can be assigned to x, y, z in 6 ways)
1+1+3=5 (1,1,3 can be assigned to x, y, z in 3 ways)
1+2+2=5 (1,2,2 can be assigned to x, y, z in 3 ways)
So total 21 ways.
Hence, the number of possibilities of x+y+z+u+t=20 has to be a multiple of 21
In the options, only B is a multiple of 21.
There can be 21 cases satisfying this equation.
0+0+5=5 (0,0,5 can be assigned to x, y, z in 3 ways)
0+1+4=5 (0,1,4 can be assigned to x, y, z in 6 ways)
0+2+3=5 (0,2,5 can be assigned to x, y, z in 6 ways)
1+1+3=5 (1,1,3 can be assigned to x, y, z in 3 ways)
1+2+2=5 (1,2,2 can be assigned to x, y, z in 3 ways)
So total 21 ways.
Hence, the number of possibilities of x+y+z+u+t=20 has to be a multiple of 21
In the options, only B is a multiple of 21.
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Find the number of non-negative integral solutions to the system of eq...
Solution:
Given equations are:
x y z u t = 20 and x y z = 5
Number of non-negative integral solutions to the system of equations can be found using generating functions as follows:
Generating function for x, y, z, u, t can be written as:
(1 + x + x2 + x3 + …)(1 + y + y2 + y3 + …)(1 + z + z2 + z3 + …)(1 + u + u2 + u3 + …)(1 + t + t2 + t3 + …)
Multiplying the above equation, we get
(1 + x + x2 + x3 + …)(1 + y + y2 + y3 + …)(1 + z + z2 + z3 + …) = (1 + x + x2 + x3 + …)5 / (1 + x + x2 + x3 + …)2
Using the formula for the sum of geometric progression, we get
(1 + x + x2 + x3 + …)k = 1 / (1 - x)k, where k = 2 or 5
Substituting the above formula in the generating function, we get
(1 + x + x2 + x3 + …)5 / (1 + x + x2 + x3 + …)2 = (1 / (1 - x)5) / (1 / (1 - x)2) = (1 - x)3 / (1 - x)5 = (1 - x)–2(1 + x + x2) = ∑ (n+2)C2 xn,
where n is the number of non-negative integral solutions to the system of equations.
Now, we need to find the coefficient of x3 in the above equation, which is (5+2)C2 = 21C2 = 21*10/2 = 105.
Hence, the number of non-negative integral solutions to the system of equations x y z u t = 20 and x y z = 5 is 105.
Therefore, the correct option is (B) 336.
Given equations are:
x y z u t = 20 and x y z = 5
Number of non-negative integral solutions to the system of equations can be found using generating functions as follows:
Generating function for x, y, z, u, t can be written as:
(1 + x + x2 + x3 + …)(1 + y + y2 + y3 + …)(1 + z + z2 + z3 + …)(1 + u + u2 + u3 + …)(1 + t + t2 + t3 + …)
Multiplying the above equation, we get
(1 + x + x2 + x3 + …)(1 + y + y2 + y3 + …)(1 + z + z2 + z3 + …) = (1 + x + x2 + x3 + …)5 / (1 + x + x2 + x3 + …)2
Using the formula for the sum of geometric progression, we get
(1 + x + x2 + x3 + …)k = 1 / (1 - x)k, where k = 2 or 5
Substituting the above formula in the generating function, we get
(1 + x + x2 + x3 + …)5 / (1 + x + x2 + x3 + …)2 = (1 / (1 - x)5) / (1 / (1 - x)2) = (1 - x)3 / (1 - x)5 = (1 - x)–2(1 + x + x2) = ∑ (n+2)C2 xn,
where n is the number of non-negative integral solutions to the system of equations.
Now, we need to find the coefficient of x3 in the above equation, which is (5+2)C2 = 21C2 = 21*10/2 = 105.
Hence, the number of non-negative integral solutions to the system of equations x y z u t = 20 and x y z = 5 is 105.
Therefore, the correct option is (B) 336.
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