Can you please check my work and help me with the parts that I don't understand?
For each event, choose the most appropriate term and solve the problem.
1. In a sweepstakes with nine hundred entries, the first winner selected receives the grand prize, the second receives first prize, and so on until all thirty prizes are awarded. How many possible outcomes are there?
900^ P 30
Counting Principle, combination, factorial, or permutation
I chose permutation- My work: 900!/(900 - 30) = 900!/870!.. From here, do I simplify?
2. Six friends go to a movie. How many ways can they sit in a row of six seats?
Counting Principle, combination, factorial, or permutation. I chose permutation
My work: 6x5x4x3x2x1 = 720 ways
3.Of the fifty states, five are randomly selected to have their governor participate in a summit. How many different groups of governors can go? 50!/(50 - 5)!.5!
Counting Principle, combination, factorial, or permutation
I chose combination- I'm really not sure how to solve this.
Thank you so much for your help!
#1 correct. 900P30 is the simplest possible answer. Trying to evaluate it does not add much to understanding the value of the answer. Suffice it to say that 2.6*10^88 is pretty big.
#2 ok
#3 ok 50C5 - combinations, since the order of their selection does not matter.
Thanks :)
If Jeremy has 17 chocolate bars and he give 3 to Lucy and gets 5 from Bruno, how many chocolate bars does Jeremy have now? All of my friends say "diabetes" but I have repeatedly told them that diabetes is not a number but they stick with it. I believe that Jeremy has 19 chocolate bars left and that's what they wrote down (my friends)but they said it was just to trick cheaters. I for one, believe that the answer is 19.
Let's go through each question and check your work:
1. You correctly identified that this problem involves finding the number of possible outcomes. Since the order of the prizes matters, we need to use the concept of permutations. You started correctly by using the formula for permutations: n! / (n - r)!, where n is the total number of items (in this case, entries) and r is the number of items selected (in this case, prizes awarded). However, the calculation seems to be incorrect. It should be 900! / (900 - 30)! = 900! / 870!. So far, your work is correct. To further simplify this expression, you can cancel out common factors. In this case, you can cancel out all the terms from 900 down to 871, leaving you with the expression 900 x 899 x 898 x ... x 871. At this point, you can stop simplifying and leave it as it is.
2. For the question about the friends sitting in a row of six seats, you correctly recognized that this involves finding the number of arrangements or permutations. There are 6 friends, so you multiply the number of choices for each seat: 6 x 5 x 4 x 3 x 2 x 1 = 720 ways. Your work is correct.
3. In the third question, you correctly identified that this is a combination problem. The formula for combinations is n! / (r! * (n - r)!), where n is the total number of items (in this case, states) and r is the number of items selected (in this case, groups of governors). However, there seems to be an error in your calculation. It should be 50! / ((50 - 5)! * 5!) = 50! / (45! * 5!). The factorial notation "!" means multiplying all the numbers from the chosen number down to 1. To simplify this expression, you can cancel out common factors. In this case, you can cancel out all the terms from 50 down to 46, leaving you with the expression 50 x 49 x 48 x 47. Then, you also have the 5! in the denominator, which is 5 x 4 x 3 x 2 x 1. Again, you can stop simplifying at this point.
Overall, your approach and understanding of permutations and combinations are correct. Just be careful with the calculations, and make sure you simplify as much as possible before stopping. Great job!