Example: Find the total number of ways to order a dinner when selecting one entree from 10 options, one drink from 5 options, and one dessert from 6 options.
Answer: 10 entrees * 5 drinks * 6 dessert = 300 different dinners
The addition counting principle indicated that the sums from events can be grouped in such a way that the groupings do not share possible outcomes, then the total number of ways each group can happen should then be added together.
Example: Find the total number of ways a 3 symbol code made of digits or letters can be made if at least one symbol must be a letter.
Answer: There are 26 different letters and 10 different digits
1-letter and 2-digit code (A # #): 26 * 10 * 10 = 2600 A## codes.
But, the letter does not have to be in the first spot.
It can occur in any of the 3 spots, so we multiply the total by 3.
26o0 * 3 = 7800 different 1-letter 2-digit codes
2-letters and 1 digit code (A A #): 26 * 26 * 10 = 6760 AA# codes
But, the digit does not have to be in last spot.
It can occur in ay of the 3 spots, so we multiply the total by 3.
6760 * 3 = 20,280 different 2-letter 1-digit codes
3-letters (A A A): 26 * 26* 26 = 17,576 AAA codes
Now, add together the total from each group:
7800 + 20,280 + 17576 = 45,656 different 3-symbol codes
(with at least 1 letter)
There are some typical modifications to the above problem, such as limiting the digits to only evens (5 options), or not allowing a symbol to repeat. For the latter modification, the number of symbols available decreases for one that is already used.
Example: How many 3 letter codes can be made if no letter can be repeated?
Answer: 3-letters (A A A): 26 * 25 * 24 = 15,600 different 3-letter codes
(with no repeated letters)
There will be a quiz on Friday covering the lessons from this week: counting principles, permutations (Tuesday), and combinations (Wednesday).
Homework: Page 340, #2 - 10 even, #11; Page 341, #2 - 8a even (omit 8b)
No comments:
Post a Comment