Probability and Counting: Passwords, Combinations, and Geometry, Study notes of Cryptography and System Security

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PROBABILITY

Counting

Multiple Stages With replacement

• HOW MANY PASSWORD ARE POSSIBLE STARTING WITH TWO LETTERS FOLLOWED BY FOUR NUMERALS?
• (26)(26)(10)(10)(10)(10) = 6,760,

WITHOUT REPLACEMENT

• How many three letter words from { a,b,c,d,e} without using a letter more than once?

With restrictions

• How many three letter words from { a,b,c,d,e} without using a letter more than once?

• How many three letter words from { a,b,c,d,e} without using a letter more than once and a vowel in the middle?
• Ans. (4)(2)(3) = 24

• How many three digit numbers have one zero?
• Case 1 zero in the middle (9)(1)(9) = 81
• Case 2 zero on the right (9)(9)(1) = 81
• Total 81 +81 =

Complementary

(finding opposite first)

• How many ways to seat four boys and three girls in a row so that at least two boys sit together?

Multiple methods

• In how many ways may 3 officers (Pres. ,V.P. , Sec) be picked from a 12 member club, If Al and Bob won’t serve together?

• In how many ways may 3 officers (Pres. ,V.P. , Sec) be picked from a 12 member club, If Al and Bob won’t serve together?

• Ans. Ways to pick three of 12: (12)(11)(10)

• Ways to pick three with Al and Bob: (3)(2)(10) =60. Thus 1320-60 = 1260 ways to pick without both.

• How many ways tho arrange the letters of Mississippi?
• Ans. There are 11! ways to arrange the letters. There are 4! ways to arrange the “s” and “I”, and 2! For “p”. Thus 11!/[(4!)(4!)(2!) =34650 diferent arrangements.

Symmetry

• How many ways can 6 people be seated at a round table. It does not matter who is seated north. What matters is who is on a persons left and right.

Combinations

(order does not matter)

• In How many ways can a committee of four be picked from a club of ten.

• In How many ways can a committee of four be picked from a club of ten.
• Ans. There are (10)(9)(8)(7) way to pick, but there are 4! Ways to arrange each pick so there are duplications. Thus there are (10)(9)(8)(7)/4! Different committees. Combinations are represented by 10 C4.