If ${ }^{2 n} C_{r} = { }^{2 n} C_{r+2}$,find $r$.

How many $3$-digit numbers can be formed using $0, 1, 3, 5, 7$ when repetition is allowed?

How many words can be made from the letters of the word $BHARAT$ in which $B$ and $H$ never come together?

All the five-digit numbers in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The $97^{th}$ number in the list does not contain the digit:

The total number of natural numbers of six digits that can be made with digits $1, 2, 3, 4$,if all digits are to appear in the same number at least once,is

$A$ company has $10$ employees. The company has decided to form a team including at least three employees and also excluding at least three employees. Then the number of ways of forming the team is