How many strings are there using the letters of the word INTERMEDIATE, if the vowels and consonants are alternative?

How many strings are there using the letters of the word INTERMEDIATE, if all the vowels are together?

How many strings are there using the letters of the word INTERMEDIATE, if vowels are never together?

How many strings are there using the letters of the word INTERMEDIATE, if no two vowels are together.

Each of the digits $1,1,2,3,3$ and $4$ is written on a separate card. The six cards are then laid out in a row to form a $6-$digit number.

How many distinct $6-$digit numbers are there?

Each of the digits $1,1,2,3,3$ and $4$ is written on a separate card. The six cards are then laid out in a row to form a $6-$digit number.

How many of these $6-$digit numbers are even?

Each of the digits $1,1,2,3,3$ and $4$ is written on a separate card. The six cards are then laid out in a row to form a $6-$digit number.

How many of these $6-$digit numbers are divisible by $4?$

If the letters of the word GARDEN are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, then find the rank of the word GARDEN?

If the letters of the word GARDEN are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, then find the rank of the word DANGER.