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Question

Total number of integral solutions of |x|+|y|+|z|=15 is k, then k41=   


Solution

Let x,y,z be positive integers.
Then number of solutions = 14C2=91
But x,y,z can be negative too.
So, total number of solutions (none of x,y,z is zero) is 91×2×2×2=728     ...(1)

Let one of x,y,z is equal to zero.
Then number of possible solutions = 14C1=14
But other two can be negative (14×2×2) and one of three can be zero.
Hence, number of solutions =14×2×2×3=168     ...(2)

Let two of x,y,z is zero, then number of solutions is 6     ...(3)

Adding (1),(2) and (3)
Total number of integral solutions, k = 902
Hence, k41=90241=22

Mathematics

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