Module 11 Homework Problems
Note that
$$\begin{align}_nC_m=\binom{n}{m}=\frac{n!}{m!\cdot (n-m)!}& &\text{and} & &_nP_m=\frac{n!}{(n-m)!} \end{align}$$.
Compute the following (please show at least some work, or that you plugged everything into a formula)
- \(_6P_3\)
- \(_2P_2\)
- \(_5P_4\)
- \(_nP_{(n-1)}\) where \(n\) is an arbitrary (unknown) natural number
- \(_0P_0\)
- \(\binom{5}{2}\)
- \(\binom{6}{3}\)
- \(\binom{10}{9}\)
- \(\binom{n}{(n-1)}\) where \(n\) is an arbitrary natural number
Solve the following problems using the multiplication principle, or permutations and combinations techniques. Do not rely on formulas alone!
- How many ways can you select a committee of seven from a group of 14 people?
- How many bitstrings with \(8\) bits are there?
- How many base 3 numbers are there with exactly 5 digits?
- How many three-letter “words” are there, so that the middle letter is a vowel and the first and third letter are consonants?
- How many ways can you rearrange the letters of the word “uncopyrightable” to form distinct (no repeated) words?
- How many unique (non-repeated) ways can you rearrange the letters of the word “sassafrass” to form “new words”? Be sure you omit duplicate words!
- Suppose you have five indistinguishable pigs that you need to sort into pens. You have three pens in which you can place pigs. How many different ways can you put five pigs into these three pens (without consideration for which specific pigs are in each of the pens)
- (Hint: Let “p” represent a pig, and let “|” represent a “divider” between pens. That way, we can represent a configuration of pigs in pens as “p | p p | p p” or “| p | p p p p”. This is a permutation problem similar to #15 or examples 4.1, 4.2 in the permutations lesson. Think of the above string of characters as a “word”)
- I walk into an animal shelter and decide I will be leaving with three cats and two dogs. The shelter is home to 10 cats and 4 dogs. How many different combinations of pets can I leave with?
- In a standard deck of 52 playing cards, how many ways can I draw a hand of five cards, where exactly three cards are spades?
- In a standard deck of 52 playing cards, how many ways can I draw a hand of seven cards with four or more diamonds?
- In a standard deck of 52 playing cards, how many ways can I draw a hand of six cards with five or more queens? (Be careful!)
21. Compute the values of each of the cells in the following table (you can use a computer or calculator; no need to show work)
| \(\binom{0}{0}\) | ||||
| \(\binom{1}{0}\) | \(\binom{1}{1}\) | |||
| \(\binom{2}{0}\) | \(\binom{2}{1}\) | \(\binom{2}{2}\) | ||
| \(\binom{3}{0}\) | \(\binom{3}{1}\) | \(\binom{3}{2}\) | \(\binom{3}{3}\) | |
| \(\binom{4}{0}\) | \(\binom{4}{1}\) | \(\binom{4}{2}\) | \(\binom{4}{3}\) | \(\binom{4}{4}\) |
22. Look up Pascal’s triangle and how one can generate it. Use it to generate the next row of the table above.
23. (5 pts for each of the following)
a.) Expand \((x+y)^0\) (try to represent your answer with \(x\) and \(y\) with zero’s out in front of each)
b.) Expand \((x+y)^1\) (try to represent your answer \(1\)’s in front of \(x\) and \(y\))
c.) Expand \((x+y)^2\)
d.) Expand \((x+y)^3\)
e.) Expand \((x+y)^4\)
These expressions above are called “(powers of) binomials.”
24. Describe the relationship between Pascal’s triangle and the coefficients in front of each term in (a)-(e) above.
This relationship is why \(\binom{n}{m}\) is called a “binomial coefficient.”
25. Use your observation to expand the expressions \((x+y)^5\) and \((x+y)^6\) WITHOUT FOILing it all out by hand.
26.(Bonus +5) Use Pascal’s triangle to expand the expression \((x+1)^5\) (somehow indicate in your work that you indeed used Pascal’s triangle, or explain how you used it in words)
27. (Bonus +10) Use Pascal’s triangle to expand the expression \((x+2)^6\) (somehow indicate in your work that you indeed used Pascal’s triangle, or explain how you used it in words)