In a group of $180$ people, there are $88$ males, nine of whom are left-handed, and there are $85$ females who are not left-handed. If six people are selected randomly from the group, find the probability that exactly four of them are left-handed or female.

A small library holds $1240$ books: $312$ of the $478$ novels $\left(N\right)$ have hard covers $\left(H\right),$ and there are $440$ books that do not have hard covers. Some of this information is shown in the Venn diagram opposite.

Find the value of $a,$of $b$ and of $c.$

A small library holds $1240$ books: $312$ of the $478$ novels $\left(N\right)$ have hard covers $\left(H\right),$ and there are $440$ books that do not have hard covers. Some of this information is shown in the Venn diagram opposite.

A random selection of $25$ of these books is to be donated to a charity group. The charity group hopes that at least $22$ of the books will be novels or hard covers. Calculate the probability that the charity group gets what they hope for.

A photograph is to be taken of a pasta dish and $n$ pizzas. The items are arranged in a line in random order. Event $X$ is 'the pasta dish is between two pizzas'.

Investigate the value of $P\left(X\right)$ for values of $n$ from $2$ to $5.$

A photograph is to be taken of a pasta dish and $n$ pizzas. The items are arranged in a line in random order. Event $X$ is 'the pasta dish is between two pizzas'.

Hence, express the value of $\frac{P\left(X\text{'}\right)}{P\left(X\right)}$ in terms of $n.$ Can you justify your answer for any value of $n⩾2 ?$ 

The word MARMALADE contains four vowels and five consonants. Find the number of possible arrangements of its nine letters if:

there are no restrictions on the order

The word MARMALADE contains four vowels and five consonants. Find the number of possible arrangements of its nine letters if:

the arrangement must begin with the four vowels.