Explanation
Step 1: Formulate the Equation
Let x1,x2,x3,x4 be the marks obtained in the four papers. We need to find the number of non-negative integer solutions to:
Subject to the constraint:
Step 2: Use Generating Functions
The number of ways is equal to the coefficient of x2m in the expansion of:
Using the formula for a finite geometric series, 1+x+⋯+xm=1−x1−xm+1, we get:
G(x)=(1−x1−xm+1)4=(1−xm+1)4(1−x)−4
Step 3: Expand the Terms
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Using Binomial Theorem for (1−xm+1)4:
(1−xm+1)4=(04)−(14)xm+1+(24)x2m+2−…
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Using Negative Binomial Theorem for (1−x)−4:
(1−x)−4=r=0∑∞(4−1r+4−1)xr=r=0∑∞(3r+3)xr
Step 4: Find the Coefficient of x2m
We look for terms in the product that result in x2m:
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Term 1: (04)×(coefficient of x2m in ∑(3r+3)xr)
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Term 2: −(14)xm+1×(coefficient of x2m−(m+1) in ∑(3r+3)xr)
=−4×(3(m−1)+3)=−4(3m+2)
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The next term in the first expansion is x2m+2, which is already higher than x2m, so no further terms contribute.
Step 5: Final Calculation
Total ways N=(32m+3)−4(3m+2)
Expanding the combinations:
N=6(2m+3)(2m+2)(2m+1)−4⋅6(m+2)(m+1)m
N=62(m+1)[(2m+3)(2m+1)−2m(m+2)]
N=3m+1[(4m2+8m+3)−(2m2+4m)]
Alternatively, this is often expressed in its expanded form:
Final Answer:
The number of ways is 31(m+1)(2m2+4m+3).
