Arrangements with alternating patterns

7

1. Find the number of different arrangements of the 9 letters in the word ANDROMEDA in which no consonant is next to another consonant. (The letters D, M, N and R are consonants and the letters A, E and O are not consonants.)
2. Find the number of different arrangements of the 9 letters in the word ANDROMEDA in which there is an A at each end and the Ds are not together.
   Four letters are selected at random from the 9 letters in the word ANDROMEDA.
3. Find the probability that this selection contains at least one D and exactly one A .
   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 Find the number of different ways in which all 8 letters of the word TANZANIA can be arranged so that

1. all the letters A are together,
2. the first letter is a consonant ( \(\mathrm { T } , \mathrm { N } , \mathrm { Z }\) ), the second letter is a vowel ( \(\mathrm { A } , \mathrm { I }\) ), the third letter is a consonant, the fourth letter is a vowel, and so on alternately. 4 of the 8 letters of the word TANZANIA are selected. How many possible selections contain
3. exactly 1 N and 1 A ,
4. exactly 1 N ?

7 Find the number of different ways in which all 9 letters of the word MINCEMEAT can be arranged in each of the following cases.

1. There are no restrictions.
2. No vowel (A, E, I are vowels) is next to another vowel.
   5 of the 9 letters of the word MINCEMEAT are selected.
3. Find the number of possible selections which contain exactly 1 M and exactly 1 E .
4. Find the number of possible selections which contain at least 1 M and at least 1 E .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 Find the number of ways the 9 letters of the word SEVENTEEN can be arranged in each of the following cases.

1. One of the letter Es is in the centre with 4 letters on either side.
2. No E is next to another E.
   5 letters are chosen from the 9 letters of the word SEVENTEEN.
3. Find the number of possible selections which contain exactly 2 Es and exactly 2 Ns.
4. Find the number of possible selections which contain at least 2 Es.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 The word ARGENTINA includes the four consonants R, G, N, T and the three vowels A, E, I.

1. Find the number of different arrangements using all nine letters.
2. How many of these arrangements have a consonant at the beginning, then a vowel, then another consonant, and so on alternately?

5

1. Find the number of different ways of arranging all nine letters of the word PINEAPPLE if no vowel (A, E, I) is next to another vowel.
2. A certain country has a cricket squad of 16 people, consisting of 7 batsmen, 5 bowlers, 2 allrounders and 2 wicket-keepers. The manager chooses a team of 11 players consisting of 5 batsmen, 4 bowlers, 1 all-rounder and 1 wicket-keeper.
   1. Find the number of different teams the manager can choose.
   2. Find the number of different teams the manager can choose if one particular batsman refuses to be in the team when one particular bowler is in the team.

6 A test consists of 4 algebra questions, A, B, C and D, and 4 geometry questions, G, H, I and J.
The examiner plans to arrange all 8 questions in a random order, regardless of topic.

1. (a) How many different arrangements are possible?
   (b) Find the probability that no two Algebra questions are next to each other and no two Geometry questions are next to each other. Later, the examiner decides that the questions should be arranged in two sections, Algebra followed by Geometry, with the questions in each section arranged in a random order.
2. (a) How many different arrangements are possible?
   (b) Find the probability that questions A and H are next to each other.
   (c) Find the probability that questions B and J are separated by more than four other questions.

3. The letters of the word CHAFFINCH are written on cards.
i. In how many ways can the letters be rearranged with no restrictions.
ii. In how many difference ways can the letters be rearranged if the vowels are to have at least one consonant between them.
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6. The 20 members of a club consist of 10 Town members and 10 Country members.

1. All 20 members are arranged randomly in a straight line. Determine the probability that the Town members and the Country members alternate.
2. Ten members of the club are chosen at random. Show that the probability that the number of Town members chosen is no more than \(r\), where \(0 \leqslant r \leqslant 10\), is given by
   \(\frac { 1 } { N } \sum _ { i = 0 } ^ { r } \left( { } ^ { 10 } C _ { i } \right) ^ { 2 }\)
   where \(N\) is an integer to be determined.
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