Permutations and Combinations Questions

In a school sports club, $25$ students decided to participate in a table tennis singles tournament. Each time a student loses a match, they are eliminated from the tournament. What is the minimum number of matches required to determine the winner?

How many members should at least be there in a Club so that it is guaranteed that at least two members have the same month of birth?

A clerk is given 3 letters written in a language unknown to her. She is also given an envelope with an address corresponding to each letter written in the same language. If the clerk puts the letters in the envelopes, in how many ways can she do it so that none of the 3 envelopes contains the letter with the correct address?

How many ways a captain and a vice-captain can be selected from a team of 11 players?

A vendor has $9$ different fruits. In how many ways can he select $3$ fruits?

A committee of $3$ members is to be formed from $5$ men and $4women$. What is the probability that the committee contains at least one woman?

In a school sports club, 25 students decided to participate in a table tennis singles tournament. Each time a student loses a match, they are eliminated from the tournament. What is the minimum number of matches required to determine the winner?

What is the probability of drawing an ace from a pack of 52 cards?

From a group of $7\text{men}$ and $6\text{women}$, five persons are to be selected to form a committee so that at least $3\text{men}$ are there in the committee. In how many ways can it be done?

How many ways can a captain and a vice-captain be selected from a team of $11$ players?