How many ways are there to make different words (not necessarily meaningful) by changing order of letters in the word $b_1b_1...b_1b_2b_2...b_2...b_mb_m...b_m$, where $b_1$ appears $k_1$ times, $b_2$ appears $k_2$ times, ... , $b_m$ appears $k_m$ times.
My answer is $$\sum_{k_1+k_2+...+k_m = n \\ 0\leq k_i \leq n}\quad\Large\frac{n!}{k_1!k_2!...k_m!}b_1^{k_1}b_2^{k_2}...b_m^{k_m}$$
Is this correct?
Your answer of $60$ corresponds to $\large\frac{6!}{1!2!3!}$, the first part of your formula, which represents the number of permutations of a $n$ letter word with $k_1$ of the first kind, $k_2$ of the second kind,.... $\Sigma k_i = n$
This represents the multinomial coefficient
What you have possibly mixed it up with is the Multinomial Theorem, which states
If events $E_1, E_2, ... E_k$ can occur with probabilities $p_1, p_2, ... p_k$ respectively, then the probability that events $E_1, E_2, ... E_k$ will occur $X_1, X_2, ... X_k$ times respectively is
$$\Large\frac{N!}{X_1!X_2! ... X_k!}p_1^{X_1}p_2^{X_2}... p_k^{X_k}$$
where $X_1 + X_2 + ... X_k = N$