these counting formula really confuses, I don’t when to use what.
First, the multinomial and binomial formulae aren’t for counting; they’re for computing probabilities for those particular distributions.
Second, a multinomial distribution isn’t part of the CFA curriculum; I don’t know where you heard the term, but put it out of your mind.
The only counting formulae in the CFA curriculum are those for permutations and combinations.
You use the permutation formula when the order of the selections matters; you use the combination formula when the order doesn’t matter.
Multinomial formula is in reading 8, and there are questions on it in the EOC.
The answer is close to 13 billion. We can label any of 18 funds high risk (the first slot), then any of 17 remaining funds, then any of 16 remaining funds, then any of 15 remaining funds (now we have 4 funds in the high risk group); then we can label any of 14 remaining funds above-average risk, then any of 13 remaining funds, and so forth. There are 18! possible sequences. However, order of assignment within a category does not matter. For example, whether a fund occupies the first or third slot of the four funds labeled high risk, the fund has the same label (high risk). Thus there are 4! ways to assign a given group of four funds to the four high risk slots. Making the same argument for the other categories, in total there are (4!)(4!)(3!)(4!)(3!) equivalent sequences. To eliminate such redundancies from the 18! total, we divide 18! by (4!)(4!)(3!)(4!)(3!). We have 18!/(4!)(4!)(3!)(4!)(3!) = 18!/(24)(24)(6)(24)(6) = 12,864,852,000. This procedure generalizes as follows.
- Multinomial Formula (General Formula for Labeling Problems). The number of ways that n objects can be labeled with k different labels, with n_{1} of the first type, n_{2} of the second type, and so on, with n_{1} + n_{2} + … + n_{k} = n, is given by