Combinatrics/4 letter code word

Question

Question reads: How many distinct $4$-letter code words can be made from the letters in the words "Pizza Pie" if the first letter must be a vowel and second must be a consonant. Answer is 98

I understand how to come up with the $4$ letter code words using the permutations and combinations, but I am being thrown off by those first two spots and how to incorporate them.

Any help would be greatly appreciated. Thanks!!

Answer 1

You have two cases to check for.

Case 1: You use either A or E as the first letter. Case 2: You use I as the first letter.

Next, you choose either a P or a Z as the consonant for the second letter.

This means for case 1, you are left with 4 distinct letters with multiplicities 1,1,2,2.

For case 2, you are left with 5 distinct letters with multiplicities 1,1,1,1,2.

Using the sum principle, the total number of distinct words is given by:

$$\begin{align*}& (\text{# of ways to choose A or E})\cdot (\text{# of ways to choose consonant})\cdot P_1 \\ + & (\text{# of ways to choose the letter I})\cdot (\text{# of ways to choose consonant})\cdot P_2 \\ = & 2\cdot 2\cdot P_1 + 1\cdot 2\cdot P_2\end{align*}$$

where $P_1$ is the number of ways to choose the last two letters given multiplicities 1,1,2,2 and $P_2$ is the number of ways to choose the last two letters given multiplicities 1,1,1,1,2.

In this case, $P_1 = 2+4\cdot 3$ and $P_2 = 1+5\cdot 4$, making the total number of distinct words $98$.

Answer 2

HINT:

You have $3$ vovels and $2$ consonants, so you can choose the first $2$ letters in $$ 3\cdot 2 $$ different ways.

For the rest of the letters you sadly have to use cases

1) You picked $A$ as your first letter (vovel) and $Z$ as your second letter (consosnant), then you have $2$ consonants and $2$ vovels left to arrange the last two letters.

2) You picked $A$ as your first letter (vovel) and $P$ as your second letter (consosnant), then..

Can you do the rest?

Hope this helped :)