4. Using the letters of the word INTERMITTENT how many...

4. Using the letters of the word INTERMITTENT how many 12-letter arrangements can be formed: without restrictions (2 marks) if each word begins with T and ends with N (2 marks) if the 4 T's are together (2 marks) if each word begins with 2 T's (2 marks) if each word begins with exactly 2 T's (2 marks)

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Transcript

00:01 Hello, so to solve these problems, we need to consider the nature of permutations and the specific restrictions given for each scenario.

00:06 The word intermittent has 12 letters with certain repetitions, 4 ts, 2 ns, and 2 is.

00:13 So, first we're looking without restrictions.

00:17 This is the number of ways to arrange the 12 letters where there are duplicates.

00:20 The formula for this is 12 factorial over 4 factorial, 2 factorial times 2 factorial, where 12 factorial is the factorial of 12, and the denominators are the factorials of the counts of each repeated letter.

00:37 So for this, we get a total of approximately 4 ,989 ,600 arrangements.

00:47 Now moving on to each word begins with t and ends with n as the restriction.

00:51 In this case, we fix t at the beginning and n at the end.

00:54 This leaves us with 10 letters to arrange, 3ts, 1n, and the rest of the letters.

00:58 The formula becomes 10 factorial over 3 factorial, 2 factorial, 1 factorial.

01:06 And so we get approximately, or actually exactly 302 ,400 arrangements.

01:15 And now we look at 4ts together.

01:19 Treat the 4ts as a single entity.

01:21 This gives us 9 entities to arrange, the 4ts together and the other 8 individual letters.

01:25 The arrangement is 9 factorial.

01:27 However, within the four ts, they can be arranged in four factorials...

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