The answer to above problem is c*(n-1)+1. Because the answer is strictly greater than c*(n-1) as the combination c11,c12,...,c1n;c21,c22,...,c2n;...;c(n-1)1,c(n-1)2,...,c(n-1)n (where cij is color j) is a possible outcome and contains no n-group of any color. Now, let us assume that we took out c*(n-1) socks and still there is no n-group than I claim that the outcome is a permutation of above mentioned outcome.
Proof of claim:
let ni, for i=1,2,3,...c be number of socks there are which have color j. And assume that there is no n-group.
Then:
for all i, ni<=n-1
summing up for all i:
sum(ni)<=c(n-1)
Now let us suppose that color I does not have n-1 number, then the sum(ni)
The answer to above problem is c*(n-1)+1. Because the answer is strictly greater than c*(n-1) as the combination c11,c12,...,c1n;c21,c22,...,c2n;...;c(n-1)1,c(n-1)2,...,c(n-1)n (where cij is color j) is a possible outcome and contains no n-group of any color. Now, let us assume that we took out c*(n-1) socks and still there is no n-group than I claim that the outcome is a permutation of above mentioned outcome.
Proof of claim:
let ni, for i=1,2,3,...c be number of socks there are which have color j. And assume that there is no n-group.
Then:
for all i, ni<=n-1
summing up for all i:
sum(ni)<=c(n-1)
Now let us suppose that color I does not have n-1 number, then the sum(ni)