AI Made - Week 10 - 11

• What is the power of counting?

  Determining arrangements or selections of values
• What are counting techniques used for?

  To determine possible outcomes for events
• What are applications of counting principles?

  • Probability and Statistics
  • Algorithms
  • Cryptography
  • Decision Making
• What is combinatorics?

  A branch of mathematics about counting and arranging
• What are the fundamental principles of combinatorics?

  • Sum Rule
  • Product Rule
• What does the sum rule state?

  If m and n are mutually exclusive, then m + n
• How many ways can a customer order from 3 vegetarian and 5 non-vegetarian courses?

  8 ways
• How many outfit options are there with 15 jeans and 20 t-shirts?

  35 ways
• What does the product rule state?

  If m and n are independent, then m × n
• How many passwords can be created with two lowercase letters and two digits?

  67,600 ways
• How many passwords can be created with two lowercase letters and two digits without repetition?

  58,500 ways
• What is the difference between arrangement and selection?

  • Arrangement: Order matters (permutation)
  • Selection: Order does not matter (combination)
• What is factorial?

  Product of all integers from 1 to n
• What is the notation for factorial?

  n!
• What is the value of 0! and 1!?

  Both are equivalent to 1
• What is the value of 4!?

  24
• What is the value of 7!?

  5,040
• What is the value of 8! / 5!?

  336
• How many ways can 5 persons be seated in 5 chairs?

  120 ways
• How many 4-digit postal codes can be formed with no repeated digits?

  5,040 postal codes
• How many ways can 5 people line up if 2 must stand next to each other?

  48 ways
• What is permutation?

  Different arrangements of objects in order
• What are the categories of permutation?

  • Linear
  • Circular
• What is linear permutation?

  Arrangement of objects in a line or sequence
• What is the notation for linear permutation?

  $\frac{n!}{(n - r)!}$
• How many ways can 6 persons be seated around a circular table?

  120 ways
• What is circular permutation?

  Arrangement of objects in a circle
• How many ways can 8 beads be arranged in a bracelet?

  5,040 ways
• How many ways can 6 ladies sit in a circular table with 3 beside each other?

  36 ways
• How many arrangements of necklaces can be made with 7 distinct charms?

  360 ways
• What is the formula for permutations with alike objects?

  $\frac{n!}{r_1! r_2! \ldots r_k!}$
• How many permutations are in the word TAGAYTAY?

  1,680 ways
• How many different outcomes are there for 4 heads and 2 tails in 6 coin tosses?

  15 ways
• What is a combination?

  • Selection of objects where order does not matter
• What is the notation for combination?

  $\frac{n!}{r!(n - r)!}$
• How many ways can a committee of four be selected from 11 persons?

  330 ways
• How many ways can a committee of four be selected with at least 3 doctors?

  65 ways
• How many ways can 3 doctors and 1 nurse be selected?

  10 ways
• How many ways can 4 doctors be selected?

  5 ways
• What is the total number of ways to select the committee?

  • 10 ways (3 doctors, 1 nurse)
  • 5 ways (4 doctors)
  • Total: 65 ways