# Discrete Math Counting

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How many permutations of the letters ABCDEFGH contain

a. the string ED?

b. the string CDE?

c. the strings BA and FGH?

d. the strings AB, DE and GH?

e. the strings CAB and BED?

f. the strings BCA and ABF?

How many ways are there for eight men and five women to stand in a line so that no two women stand next to each other? [Hint: First position the men and then consider possible positions for the women.]

A coin is flipped 10 times where each flip comes up either heads or tails. How many possible outcomes

a. are there in total?

b. contain exactly two heads?

c. contain at most three tails?

d. contain the same number of heads and tails?

How many positive integers between 50 and 100

a. are divisible by 7? Which integers are these?

b. are divisible by 11? Which integers are these?

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##### Solution Summary

The discrete math counting is examined. How permutations of the letters ABCDEFGH contain are determined.

##### Solution Preview

Solution to Q1:

(a) To ensure that we can have the string "ED", we consider "ED" as a whole. So, we can permutate A, B, C, ED, F, G, H. Hence, there are 7!=5040 permutations of the letters ABCDEFGH that contain the string ED.

(b) To ensure that we can have the string "CDE", we consider "CDE" as a whole. So, we can permutate A, B, CDE, F, G, H. Hence, there are 6!=720 permutations of the letters ABCDEFGH that contain the string CDE.

(c) To ensure that we can have the strings "BA" and "FGH", we consider each of "BA" and "FGH" as a whole. So, we can permutate BA, C, D, E, FGH. Hence, there are 5!=120 permutations of the letters ABCDEFGH that contain the strings BA and FGH.

(d) Similar to part (c), we can permutate AB, C, DE, F, GH. Hence, there are 5!=120 permutations of the letters ABCDEFGH that contain the ...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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