How many times does the digit $5$ appear when writing natural numbers from $1$ to $100$?

$6$ different letters of an alphabet are given. Words with $4$ letters are formed from these given letters. The number of words which have at least one letter repeated and no two same letters are together is:

All the letters of the word $ANIMAL$ are permuted in all possible ways and the permutations thus formed are arranged in dictionary order. If the rank of the word $ANIMAL$ is $x$,then the permutation with rank $x$,among the permutations obtained by permuting the letters of the word $PERSON$ and arranging the permutations thus formed in dictionary order is

$A$ number is called a palindrome if it reads the same backward as well as forward. For example,$285582$ is a six-digit palindrome. The number of six-digit palindromes,which are divisible by $55$,is ...... .

If $^nP_3 + ^nC_{n-2} = 14n$,then $n = $

If ${ }^{n} P_{r}={ }^{n} P_{r+1}$ and ${ }^{n} C_{r}={ }^{n} C_{r-1}$,then the value of $r$ is equal to: