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How many words can be formed using the letters of the word $INDEPENDENCE$,such that all vowels always come together?

All the letters of the word $PUBLIC$ are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word $PUBLIC$ is

Find the number of $4$-digit numbers that can be formed using the digits $1, 2, 3, 4, 5$ if no digit is repeated. How many of these will be even?

Find the value of $n$ such that $\frac{^{n}P_{4}}{^{n-1}P_{4}} = \frac{5}{3}$,where $n > 4$.

How many $8$-digit numbers can be formed such that all digits are distinct?

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