Solution:

Number of students = N

Let the numbers skipped by each student be n_{1},n_{2},n_{3},n_{4}...n_{N}

Sum found by first student = A - n_{1}

where A= ^{N.(N+1)}/_{2}

Similarly,

Sum found by second student = A - n_{2}

Sum found by Nth student = A - n_{N}

Sum of sum found by all N students = N.A- (n_{1}+n_{2}+n_{3}+...+n_{N})

(n_{1}+n_{2}+n_{3}+...+n_{N}) = A

Sum of sum found by all students = N.A-A

= A (N-1)

Average of sum found by all N students = ^{A.(N-1)}/_{N}

Average of averages found by N students = ^{A.(N-1)}/_{N . (N-1)} [ as each student had written only N-1 numbers]

Solving, we get,

Average of averages found by N students = (N+1)/2 =41

Thus, value of N is 81.

Hence (C)