1. permutation or combination: how many schedules are possible if you need to choose 5 events out of 20 for a vacation?
Is it permutation?? Answer is 1860480?
2. a. How many area codes are possible ( the 3 digit number can't start with a 0)?
504?
b. How many area codes start and end with an even number?
84?
c. How many area codes have 5 as the middle number?
56?
3. In the word "TIGARD", how many 3 letter words ( repeats not allowed)
a. start with a vowel? 40?
b. are all consonants? 24?
c. are all vowels? 0?
If repeats are allowed, how many consonant words are there for the previous question?
64?
1. NO
The order does not matter, Disney first or white mountains first
so
COMBINATION C(20,5)
c(20,5) = 20!/(5!*15!)
= 20*19*18*17*16 /(5*4*3*2)
= 15504
2a 9*10*10 = 900
2b 4 * 5 * 5
2c 9 * 1 * 10
6 letters
if start with A so 5 left
c(5,3) = 10
10 more start with I so
20 total
others similar
#3
Those are permutations since the order matters
a) start with a vowel:
2*5*4 = 40 , you were right
b) all consonants, means we only pick from 4
4*3*2 = 24 , you were right
c) all vowels, you are right with 0
since we only have 2 vowels and we need 3
d) all consonants , repetition allowed:
4*4*4 = 64 , again you were right.
1. To solve this problem, we need to use combination since the order of the schedule does not matter. The formula for combination is given by: nCr = n! / (r!(n-r)!), where n is the total number of events (20 in this case), and r is the number of events to be chosen (5 in this case).
Using the formula, we can substitute the values to find the answer: 20C5 = 20! / (5!(20-5)!) = 20! / (5! * 15!) = (20 * 19 * 18 * 17 * 16) / (5 * 4 * 3 * 2 * 1) = 15504.
Therefore, the number of possible schedules is 15504, not 1860480. So the answer is incorrect.
2a. To find the number of possible area codes, we need to consider that the first digit cannot be 0. This leaves us with 9 options (1-9) for the first digit. The second and third digits can be any number from 0-9, giving us 10 options each.
So, the total number of possible area codes is 9 * 10 * 10 = 900.
b. For area codes to start and end with an even number, the first and last digits can be 2, 4, 6, or 8. So, there are 4 options for the first digit, and again, 10 options each for the second and third digits.
Therefore, the total number of area codes starting and ending with an even number is 4 * 10 * 10 = 400.
c. For area codes with 5 as the middle number, the first and last digits can be any number from 1-9 (excluding 0), which gives us 9 options each. The middle digit must be 5.
So, the total number of area codes with 5 as the middle number is 9 * 1 * 9 = 81.
3a. To find the number of 3-letter words that start with a vowel in the word "TIGARD," we need to consider the vowels - 'I' and 'A'. For the first letter, we have 2 options. For the second and third letters, we have 6 options (excluding the vowel used for the first letter).
So, the total number of 3-letter words starting with a vowel is 2 * 6 * 6 = 72.
b. To find the number of 3-letter words that are all consonants in the word "TIGARD," we need to consider the consonants - 'T', 'G', 'R', and 'D'. For each of the three positions, we have 4 options.
So, the total number of 3-letter words consisting of all consonants is 4 * 4 * 4 = 64.
c. There are no 3-letter words consisting of all vowels in the word "TIGARD."
If repeats are allowed, for the previous question, we have the same options for each position. So, the total number of 3-letter words consisting of all consonants remains 4 * 4 * 4 = 64.