We know that if
$\big(X_1,X_2...X_k) \sim multinomial(n;p_1,p_2...p_k)$
$X_i \sim bin(n;p_i) $
Then, $var(X_i) = np_i(1-p_i)$.
But we have $cov(X_i,X_j) = -np_ip_j$.
So doesnt that imply $var(X_i) = cov(X_i,X_i) = -np_i^2$?
(Which is basically impossible because of the definition of variance.) Can someone explain to me what exactly am I not being able to see?