The number of words that can be written using all the letters of the word "$IRRATIONAL$" is

Words of length $10$ are formed using the letters $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated,and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then,$\frac{y}{9x} =$

Is $3! + 4! = 7!$?

There are $11$ distinct books on a shelf,consisting of $5$ Mathematics,$4$ Physics,and $2$ Chemistry books. In how many ways can they be arranged such that books of the same subject are always kept together?

$10$ different letters of the English alphabet are given. Out of these letters,words of $5$ letters are formed. How many words can be formed if at least one letter is repeated?

How many words can be formed from the letters of the word $COURTESY$,whose first letter is $C$ and the last letter is $Y$?