Q:

Selection of at least one vowel and one consonant

Find the number of ways of selection of atleast one vowel and one consonant from the word TRIPLE.

Attempt: 
We have $(I,E)$ and $(T,R,P,L)$
We choose $\dbinom{2}{1} \times \dbinom{4}{1}\times 2^4$
Explanation of attempt:
First choose one vowel.
Then choose one consonant.
Then the rest have 2 choices each - to be selected or not.
Why is my method incorrect? 
(The answer is $45$)

A:

You can choose one vowel or two vowels ( so j goes from 1 to 2) and take also one consonant or more so i (number of consonants) goes from 1 to 4. So we have :
$$1\le j\le 2$$
$$1\le i\le 4$$
$$P=\sum_{j=1}^2\binom 2 j \sum_{i=1}^4\binom 4 i=(\binom 2 1 +\binom 2 2 ) \sum_{i=1}^4\binom 4 i=3\sum_{i=1}^4\binom 4 i=3*15=45$$

