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1)

  • this answer is incorrect. correct answer is $^{1029}P_{10}$.. Not sure why permutation instead of combination.

Count the number of different ways that a disk jockey can play 10 songs from her station's library of 1029 songs.

For this one I did, $\binom{1029}{10}$ and my explanation was: Since there are 10 choices and 1029 songs to choose from, order does not matter.

2)

  • My logic behind this question is completely wrong and the professor marked me off big time for it. I don't know what the correct answer would be.

The call letters of a radio station consist of 3 or 4 letters. The first letter must be K or W. Count the number of different call letters.

My answer: 24*23*22*21 since there cannot 2 letters are already taken.

3 Answers 3

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For 1, order does matter, so it's a permutation. (Unless it's a country station, in which case all songs are really the same, so the answer is $1$.)

For 2, you need to count 3-letter and 4-letter combinations separately. I think the answer should be:

3-letter: $2\times 26 \times 26$

plus

4-letter $2\times 26 \times 26 \times 26$

equals

$36,504.$

since you should be allowed to use the same letter more than once (I know for a fact that there are radio stations with KXXX and KMMJ.)

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1) The correct answer is

$$P(1029,10) = 1029 \times 1028\times \cdots \times 1020$$

because order does in fact matter. The order in which the the disk jockey plays the songs does make a difference.

2) For this question, it is important to understand that ABC and ABCD are entirely different call letters. Thus, by the sum rule, your final answer is (# of 3 letter calls) + (# of 4 letter calls). Each of these can be found with the product rule.

We know that the first letter must be a K or a W, so the first term is 2. Because radio station call letters are allowed to use repeats, each subsequent term is 26. Therefore, I believe the final answer should be

$$(2\times 26\times 26)+(2\times 26 \times 26 \times 26) = 36504$$

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Question 1

Order matters when a disk jockey plays 10 songs. For instance, when I number the songs with $1,2,3,4,..., 1029$

Then $1,2, ..., 10$ and $10,9,8, .., 1$ are two different ways that a disk jockey can play $10$ songs when we can choose from $1029$ songs (and many more permutations). You counted them as twose 2 ways of choosing were the same, because you said order does not matter. Do you see why order does matter? I realise that this might not have been clear from the question. You can solve this by explaining what reasoning you used or simply giving both approaches and explaining when which case applies.

Question 2

We have $3$ OR $4$ letters. This means we must count:

words with $3$ letters + words with $4$ letters

We have $2$ possibilities for the first letter, from which we choose one. Thus we have $25$ letters left to choose from. Then $24$ and if we have $4$ letters then $23$. Thus, the correct answer is (assuming we cannot use the same letter twice):

$2*25*24 + 2*25*24*23$

If we can use the same letter twice, the correct answer is:

$2*26^2 + 2*26^3$