Variance of the Index of Coincidence

This concerns Friedman's Index of Coincidence (IC) in the
form (c/(N(N-1))) sum{n_i(n_i - 1), i = 1..c), as at
and in LANAKI at

Assuming the ciphertext alphabet has c letters that
occur (independently) with probabilities p_1,...,p_c,
the expected value of IC is c * sum(p_i^2, i=1..c).

I would like to know the formula for the variance of IC,
similarly as a function of c and p_1,...,p_c.