Arrangements with grouped categories

1

1. Find the number of different arrangements of the 8 letters in the word DECEIVED in which all three Es are together and the two Ds are together.
2. Find the number of different arrangements of the 8 letters in the word DECEIVED in which the three Es are not all together.

7

1. How many different arrangements are there of the 10 letters in the word REGENERATE?
2. How many different arrangements are there of the 10 letters in the word REGENERATE in which the 4 Es are together and the 2 Rs have exactly 3 letters in between them?
3. Find the probability that a randomly chosen arrangement of the 10 letters in the word REGENERATE is one in which the consonants ( \(\mathrm { G } , \mathrm { N } , \mathrm { R } , \mathrm { R } , \mathrm { T }\) ) and vowels ( \(\mathrm { A } , \mathrm { E } , \mathrm { E } , \mathrm { E } , \mathrm { E }\) ) alternate, so that no two consonants are next to each other and no two vowels are next to each other. [5]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A town council plans to plant 12 trees along the centre of a main road. The council buys the trees from a garden centre which has 4 different hibiscus trees, 9 different jacaranda trees and 2 different oleander trees for sale.

1. How many different selections of 12 trees can be made if there must be at least 2 of each type of tree? The council buys 4 hibiscus trees, 6 jacaranda trees and 2 oleander trees.
2. How many different arrangements of these 12 trees can be made if the hibiscus trees have to be next to each other, the jacaranda trees have to be next to each other and the oleander trees have to be next to each other?
3. How many different arrangements of these 12 trees can be made if no hibiscus tree is next to another hibiscus tree?

6

1. Find the number of ways in which all 9 letters of the word AUSTRALIA can be arranged in each of the following cases.
   1. All the vowels (A, I, U are vowels) are together.
   2. The letter T is in the central position and each end position is occupied by one of the other consonants (R, S, L).
2. Donna has 2 necklaces, 8 rings and 4 bracelets, all different. She chooses 4 pieces of jewellery. How many possible selections can she make if she chooses at least 1 necklace and at least 1 bracelet?

6

1. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the first letter is \(R\).
2. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the 3 letters G are together, both letters A are together and both letters E are together.
3. The letters G, R and T are consonants and the letters A and E are vowels. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if consonants and vowels occur alternately.
4. Find the number of different selections of 4 letters of the word AGGREGATE which contain exactly 2 Gs or exactly 3 Gs.

5

1. Find the number of different ways that the 13 letters of the word ACCOMMODATION can be arranged in a line if all the vowels (A, I, O) are next to each other.
2. There are 7 Chinese, 6 European and 4 American students at an international conference. Four of the students are to be chosen to take part in a television broadcast. Find the number of different ways the students can be chosen if at least one Chinese and at least one European student are included.

6 Find the number of ways all 10 letters of the word COPENHAGEN can be arranged so that

1. the vowels ( \(\mathrm { A } , \mathrm { E } , \mathrm { O }\) ) are together and the consonants ( \(\mathrm { C } , \mathrm { G } , \mathrm { H } , \mathrm { N } , \mathrm { P }\) ) are together, [3]
2. the Es are not next to each other. Four letters are selected from the 10 letters of the word COPENHAGEN.
3. Find the number of different selections if the four letters must contain the same number of Es and Ns with at least one of each.

4

1. Find the number of different ways that 5 boys and 6 girls can stand in a row if all the boys stand together and all the girls stand together.
2. Find the number of different ways that 5 boys and 6 girls can stand in a row if no boy stands next to another boy.

4 Mary saves her digital images on her computer in three separate folders named 'Family', 'Holiday' and 'Friends'. Her family folder contains 3 images, her holiday folder contains 4 images and her friends folder contains 8 images. All the images are different.

1. Find in how many ways she can arrange these 15 images in a row across her computer screen if she keeps the images from each folder together.
2. Find the number of different ways in which Mary can choose 6 of these images if there are 2 from each folder.
3. Find the number of different ways in which Mary can choose 6 of these images if there are at least 3 images from the friends folder and at least 1 image from each of the other two folders.

13. Six women and five men stand in a line for a photo.

1. In how many arrangements will all the men stand next to each other and all the women stand next to each other?
2. In how many arrangements will all the men be apart?