## Exercises 3.3 Exercises Set 5

###### Investigation Work

###### 1.

Finish all investigations from in-class activities.

###### Computation Based Exercises

###### 2.

Encipher the plaintext COURSE with a Vigenere cipher and the keyword MATH.

###### 3.

Encipher the plaintext FINISH with a Vigenere cipher and the keyword START.

###### 4.

Decipher the ciphertext YIAPKF WV ICALTU that was enciphered with a Vigenere cipher and the keyword GOOD.

###### 5.

The plaintext message FIREFLY is enciphered as XPZRDDF. Find the keyword.

###### 6.

Suppose you know the ciphertext OGPYKE to be a Vigenere encipherment of ORANGE. Find the keyword and use it to decrypt the message BPCLRA.

###### 7.

A bookshelf has space for exactly 12 books. In how many ways can the books be arranged on the bookshelf.

###### 8.

A pizza parlor offers a choice of 16 different toppings. How many five topping pizzas are possible?

###### 9.

How many three-letter “words” can be made from the eight letters FGHIJKLM if

- repetition of letters is allowed?
- repetition of letters is not allowed?

###### 10.

Calculate the following

- \(\displaystyle P(13,4) \)
- \(\displaystyle P(26,6)\)
- \(\displaystyle P(15,5)\)
- \(\displaystyle C(13,4)\)
- \(\displaystyle C(26,6)\)
- \(\displaystyle C(15,5)\)
- BONUS: Why does c. and e. have such a nice pattern?

###### 11.

The English alphabet contains 21 consonants and five vowels. If you pick 6 letters at random (all at once, no replacement) what is the probability of picking exactly one vowel.

###### 12.

The English alphabet contains 21 consonants and five vowels. If you pick 6 letters at random (all at once, no replacement) what is the probability of picking at least one vowel.

###### 13.

A blackjack hand consists of two cards from a standard 52 card deck. The first card is dealt face down and the second card is face up. Find the probability that

- The face down card is an ace.
- The face down card is an ace and the face up card is a spade. (Think about two cases here. Why?)
- The face down card is a spade and the face up card is a heart.
- The two cards have the same suit.
- One card is an ace and the other card is a 10, jack, queen, or king. (Blackjack! You win!)

###### 14.

We pick two letters at random from a set with eight As, one B, three Cs, six Ds, twelve Es, three Fs, and two Gs. We pick both at the same time and do not care about the order in which we picked them. Find the probability that both letters are the same.

###### 15.

We pick two letters at random from a set with six As, three Bs, five Cs, five Ds, nine Es, four Fs, and three Gs. We pick both at the same time and do not care about the order in which we picked them. Find the probability that both letters are the same.

###### 16.

We pick two letters at random from a set with five of each letter for the letters ABCDEFG. We pick both at the same time and do not care about the order in which we picked them. Find the probability that both letters are the same.

###### 17.

A bucket contains 1000 alphabet letters of which there are 200 A’s, 150 B’s, 400 C’s and 250 D’s. We pick two letters at random from the bucket (both at the same time, order does not matter). Find the probability that

- Both letters are a B
- Both letters are an A
- Both letters are identical
- One letter is a C and one letter is a D
- Neither letter is a D

###### Writing Based Exercises

###### 18.

A ciphertext from Civil War times that was discovered in a glass bottle at the Museum of the Confederacy is given in Figure 3.3.1.

A better image may be obtained from the Museum's website.

The full text of the message is given below.

SEAN WIEUSIUZH DTG CNP LBHXQK OZ BJQB FEQT XZBG JJOY TK FHR TPZWK ZVU RYSQ VOUPZXGG YEPH CK UASFKIPW PVVO JIZ HMN NVAEUD HYF DURJ BOVPA SF MVV FYYRDE LVPL MFYCIN XY FQEO NPK M OBZC FYXJFHOHT AS ETYV B OCAJDSVQU M ZTJV TPHY DAU FQTI UTDJ J DOGOAIA FLWHTHTI QLTR SEA LVLFLHFO

My replication cipher wheel pamphlet says the Confederacy often used one of the phrases below as keywords for their Vigenere ciphers. COME RETRIBUTION, MANCHESTOR BLUFF, or COMPLETE VICTORY.

Which one of these decrypts the message above?

###### 19.

Read an article about the Civil War ciphertext in Exercise 3.3.18 at and write a paragraph you learned. Click on article to activate the link.

BONUS: the decryption of the message as given in the article does not exactly match the decryption in Exercise 3.3.18. Where is it wrong?

###### 20.

Suppose there is a language with only 5 letters, ABCDE. We pick two letters at random from ciphertext EBDDDBECCCDADDCDDCDABACC. We pick both letters at the same time and order does not matter. Find the probability that both letters are the same. The counts for the ciphertext are three As, three Bs, seven Cs, nine Ds, and two Es.

###### 21.

Now let’s rearrange the ciphertext into three columns. (We’ll see why next time.)

E | B | D |

D | D | B |

E | C | C |

C | D | A |

D | D | C |

D | D | C |

D | A | B |

A | C | C |

- For each column compute the number of ways of picking two identical letters. Then add those up to get the total number of ways to pick two identical letters from the same column.
- If we pick two letters at random from the ciphertext, what is the probability of picking two identical letters from the same column?
- How many ways are there to pick two of the three columns?
- How many ways are there to pick two identical letters where one comes from the first column and one comes from the second column?
- How many ways are there to pick two identical letters where one comes from the first column and one comes from the third column?
- How many ways are there to pick two identical letters where one comes from the second column and one comes from the third column?
- If we pick two letters at random from the ciphertext, what is the probability of picking two identical letters from two different columns?
- If we pick two letters at random from the ciphertext, what is the probability of picking two identical letters from either the same column or from two different columns.

###### 22.

Explain why \(C(n,0)=1\) based on the formula for \(C(n,r)\text{.}\)

###### 23.

Compute the sum

for \(n=3,4,5\) and make a conjecture. Prove it! Hint

The Binomial Theorem may be useful.

###### Theorem 3.3.2.

Binomial Theorem: The expansion for \((x+y)^n\) for a positive integer \(n\) is given by

###### Enrichment Opportunities

###### 24.

Your professor gives you a set of 20 letters. There are 10 As and 10 Ds. Each letter is on a slip of paper which you must divide the 20 slips of paper into two sets. Your professor will then put each set of slips of paper into an unmarked bag (so that you don’t know which set is in which bag.) You must choose a slip of paper from one bag of the two bags and that will determine your grade in the class. How should you split them up to maximize your chance of getting an A? Describe how you would split them up and what your probability of getting an A would be. Would you take this offer?

###### 25.

Read the article “Origins of Cryptology: The Arab Contributions” about the history of Arab cryptology. (Ibrahim A. Al-Kadit, “Origins of Cryptology: the Arab Contributions”, *Cryptologia*, 16:2, 97-126, 1992.) Write a paragraph about what you found interesting in the article.