In an election,the number of candidates is $1$ greater than the number of persons to be elected. If a voter can vote in $254$ ways,then the number of candidates is:

If $^nC_r$ denotes the number of combinations of $n$ things taken $r$ at a time,then the expression $^nC_{r+1} + ^nC_{r-1} + 2 \times ^nC_r$ equals

$A$ dictionary is printed consisting of $7$-letter words only that can be formed using the letters of the word $CRICKET$. If the words are arranged in alphabetical order,as in an ordinary dictionary,then the number of words before the word $CRICKET$ is:

How many numbers less than $1000$ can be made from the digits $1, 2, 3, 4, 5, 6$ (repetition is not allowed)?

In how many different ways can the letters of the word '$OPTICAL$' be arranged so that the vowels always come together?

There are $7$ men and $8$ women in a club. $A$ committee is formed using these persons. What will be the number of ways to select $6$ persons such that at least one is a woman?