Arrangements with identical objects

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1. Find the number of different ways that a set of 10 different mugs can be shared between Lucy and Monica if each receives an odd number of mugs.
2. Another set consists of 6 plastic mugs each of a different design and 3 china mugs each of a different design. Find in how many ways these 9 mugs can be arranged in a row if the china mugs are all separated from each other.
3. Another set consists of 3 identical red mugs, 4 identical blue mugs and 7 identical yellow mugs. These 14 mugs are placed in a row. Find how many different arrangements of the colours are possible if the red mugs are kept together.

4 Three identical cans of cola, 2 identical cans of green tea and 2 identical cans of orange juice are arranged in a row. Calculate the number of arrangements if

1. the first and last cans in the row are the same type of drink,
2. the 3 cans of cola are all next to each other and the 2 cans of green tea are not next to each other.

6 A library contains 4 identical copies of book \(A , 2\) identical copies of book \(B\) and 5 identical copies of book \(C\). These 11 books are arranged on a shelf in the library.

1. Calculate the number of different arrangements if the end books are either both book \(A\) or both book \(B\).
2. Calculate the number of different arrangements if all the books \(A\) are next to each other and none of the books \(B\) are next to each other.

3 A staff car park at a school has 13 parking spaces in a row. There are 9 cars to be parked.

1. How many different arrangements are there for parking the 9 cars and leaving 4 empty spaces?
2. How many different arrangements are there if the 4 empty spaces are next to each other?
3. If the parking is random, find the probability that there will not be 4 empty spaces next to each other.

4 A builder is planning to build 12 houses along one side of a road. He will build 2 houses in style \(A\), 2 houses in style \(B , 3\) houses in style \(C , 4\) houses in style \(D\) and 1 house in style \(E\).

1. Find the number of possible arrangements of these 12 houses.

2. Road
   \(\square \square \square \square \square \square \square \square \square\)	\(\square \square \square\)

   The 12 houses will be in two groups of 6 (see diagram). Find the number of possible arrangements if all the houses in styles \(A\) and \(D\) are in the first group and all the houses in styles \(B , C\) and \(E\) are in the second group.
3. Four of the 12 houses will be selected for a survey. Exactly one house must be in style \(B\) and exactly one house in style \(C\). Find the number of ways in which these four houses can be selected.

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\includegraphics[max width=\textwidth, alt={}, center]{fcf7b1c6-cc76-4c84-998c-9de6a7e9bb2d-3_163_618_260_765} Pegs are to be placed in the four holes shown, one in each hole. The pegs come in different colours and pegs of the same colour are identical. Calculate how many different arrangements of coloured pegs in the four holes can be made using

1. 6 pegs, all of different colours,
2. 4 pegs consisting of 2 blue pegs, 1 orange peg and 1 yellow peg. Beryl has 12 pegs consisting of 2 red, 2 blue, 2 green, 2 orange, 2 yellow and 2 black pegs. Calculate how many different arrangements of coloured pegs in the 4 holes Beryl can make using
3. 4 different colours,
4. 3 different colours,
5. any of her 12 pegs.

6 A car park has spaces for 18 cars, arranged in a line. On one day there are 5 cars, of different makes, parked in randomly chosen positions and 13 empty spaces.

1. Find the number of possible arrangements of the 5 cars in the car park.
2. Find the probability that the 5 cars are not all next to each other.
   On another day, 12 cars of different makes are parked in the car park. 5 of these cars are red, 4 are white and 3 are black. Elizabeth selects 3 of these cars.
   [0pt]
3. Find the number of selections Elizabeth can make that include cars of at least 2 different colours. [5]

2 Twelve coins are tossed and placed in a line. Each coin can show either a head or a tail.

1. Find the number of different arrangements of heads and tails which can be obtained.
2. Find the number of different arrangements which contain 7 heads and 5 tails.