Permutations and combinations

In this question, 6-digit numbers do not start with 0.

Find how many 6-digit numbers have 6 different digits and are divisible by 2.

A committee of 10 people is to be selected from 9 students and 7 teachers.

Find the number of different committees that can be selected if the committee must have at least 4 students and at least 4 teachers.

These members stand in a straight line.

Find the number of ways that this can be done when all the runners stand together, all the swimmers stand together and all the gymnasts stand together.

Four of these members are selected for an event.

Find the number of ways that this can be done when at least one runner, at least one swimmer and at least one gymnast must be selected.

Find how many 4-digit numbers can be formed.

Find how many even 4-digit numbers can be formed.

Find how many 4-digit numbers that are divisible by 5 can be formed.

Solve the equation $(n + 1) \times {}^{n+1}\mathrm{C}_{12} = 33(n - 10) \times {}^n\mathrm{C}_{10}$.

Find the number of 6-character passwords that can be formed if there are no further restrictions.

Find the number of 6-character passwords that can be formed if the password starts and ends with a symbol.

Find the number of 6-character passwords that can be formed if the password:

• starts with either a symbol and then a number, or a number and then a symbol and
• ends with 2 letters.

There are no restrictions.

The number is greater than $700\,000$.

The number is greater than $750\,000$.

Find the number of possible arrangements if the first person and last person are both women.

Find the number of possible arrangements if all the children stand next to each other.

Find the number of different selections if no man is chosen.

Find the number of different selections if at least 2 women are chosen.

the O and A are not next to each other

the letters MATHS are together in this order.

There are no restrictions.

The group includes both Abby and Ben.

The group includes either Abby or Ben, but not both.

All 10 students are arranged in a line. How many arrangements are possible if there are exactly three students between Abby and Ben?

No girls are next to each other.

The 2 boys are not next to each other.

12 people, including Anjie and Bubay, are divided into 3 groups of 4 people. Anjie and Bubay must not be in the same group.

Find the number of ways in which the 3 groups can be selected.