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=22Mathematics

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