Practice section 7 of 10

The salad bar at a restaurant has 6 separate bowls containing lettuce, tomatoes, cucumber, radishes, spring onions and beetroot respectively. John decides to visit the salad bar and make a selection. At each bowl, he can choose to take some of the contents or not.

(i) Assuming that John takes some of the contents from at least one bowl, find how many different selections he can make.

(ii) John decides he is going to have 4 salad items, and one of them will be tomatoes. How many different selections can he make?

A group of 10 people consists of 3 married couples and 4 single men.

(a) A committee of 4 is to be formed from the 10 people.

(i) How many different committees can be formed?

(ii) How many different committees can be formed if the committee can consist of at most 1 married couple?

(iii) How many different committees can be formed if the committee consists of at least one man and at least one woman?

(b) The group sits at a round table with 10 seats.

(i) If each man sits next to his wife, how many ways can they be seated?

(ii) If one particular man is absent and the rest are allowed to sit without any restrictions, how many ways can they be seated?

(iii) If a particular couple is absent and the rest are allowed to sit without any restrictions, how many ways can they be seated?

A children’s game is played with 20 cards, consisting of 5 sets of 4 cards. Each set consists of a father, mother, daughter and son from the same family. The family names are Red, Blue, Green, Yellow and Orange. For example, the Red family cards are father Red, mother Red, daughter Red and son Red.

The 20 cards are arranged in a row.

(i) In how many different ways can the 20 cards be arranged so that the 4 cards in each family set are next to each other? [2]

(ii) In how many different ways can the cards be arranged so that all five father cards are next to each other, all four Red family cards are next to each other and all four Blue family cards are next to each other? [3]

In each of the following cases, find the number of ways of forming a group of 10 people from 15 couples if each group contains

(i) exactly 5 couples, [1]

(ii) no couples, [2]

(iii) exactly 2 couples. [3]

A student working on a coding project studies 11-digit quaternary sequences. A quaternary sequence is a sequence formed using the digits 0, 1, 2 or 3. Examples are 12030201131, 01122211100, 12321232123 and 00000000000. Find the number of ways that 11-digit quaternary sequences can be formed with

(i) no restriction, [1]

(ii) exactly four 0 s and four 2 s, [3]

(iii) at least two consecutive digits that are the same. [3]

Joshua tries to recall the 6-digit PIN number for his ATM card. Assume 000000 is even and valid. How many possible numbers can there be if he remembers that:

(i) the number starts and ends with the digit 5? [1]

(ii) the number is odd and the digits do not repeat? [2]

(iii) the digits do not repeat and there are exactly 3 odd digits? [2]