. In how many different ways can Tim, Savannah, Ian, Haley, Kaash, and
Myles line up?
2. There are 15 contestants in a boy’s cross country race. 3 ribbons will be
awarded. How many different ways could the ribbons be awarded?
3. How many different ways can the letters H D R F G T Y S be arranged?
4. There are 10 possible pizza toppings and the local pizza place. You can
have 4 toppings on a large pizza. How many combinations of pizza toppings
are possible?
5. There are 26 names in a jar. 6 names will be drawn to win a prize. How
many different combinations of names can be drawn?
6. What is the difference between a permutation and a combination?
Answers:
1. 10
2. Not sure
3. 49
4. Not sure
5. 43?
1. 1,2,3,4,5,6.
1,3,4,5,6,2.
1,4,5,6,2,3.
1,5,6,2,3,4.
1,6,2,3,4,5.
6,5,4,3,2,1.
6,4,3,2,1,5.
6,3,2,1,5,4.
6,2,1,5,4,3.
6,1,5,4,3,2.
5,4,3,2,1,6.
4,3,2,1,6,5.
3,2,1,6,5,4.
2,1,6,5,4,3.
1,6,5,4,3,2.
6,4,2,1,5,3.
5,3,1,6,4,2.
4,2,1,6,5,3.
3,1,6,5,4,3.
2,1,6,5,4,3.
1. To find the number of different ways the individuals can line up, we can use the concept of permutations. In this case, there are 6 people, so there are 6 choices for the first position, then 5 choices for the second position, 4 for the third, and so on until 1 choice for the last position. Therefore, the total number of different ways to line up is 6 x 5 x 4 x 3 x 2 x 1 = 720.
2. To find the number of different ways the ribbons can be awarded, we can use the concept of combinations. Since there are 15 contestants and 3 ribbons, we need to find the number of ways to choose 3 contestants out of 15. This can be calculated using the formula for combinations, which is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items to be selected. Plugging in the values, we get 15! / (3!(15-3)!) = 455 different ways the ribbons can be awarded.
3. To find the number of different ways the letters can be arranged, we can use the concept of permutations. Since there are 9 letters, we have 9 choices for the first position, then 8 choices for the second position, 7 for the third, and so on until 1 choice for the last position. Therefore, the total number of different ways the letters can be arranged is 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880.
4. To find the number of different combinations of pizza toppings, we can use the concept of combinations. Since there are 10 possible toppings and we want to select 4 toppings for a large pizza, we need to find the number of ways to choose 4 toppings out of 10. This can be calculated using the formula for combinations, which is nCr = n! / (r!(n-r)!). Plugging in the values, we get 10! / (4!(10-4)!) = 210 different combinations of pizza toppings.
5. To find the number of different combinations of names that can be drawn, we can use the concept of combinations. Since there are 26 names in the jar and we want to draw 6 names, we need to find the number of ways to choose 6 names out of 26. This can be calculated using the formula for combinations, which is nCr = n! / (r!(n-r)!). Plugging in the values, we get 26! / (6!(26-6)!) = 230,230 different combinations of names that can be drawn.
6. The difference between a permutation and a combination lies in the order of the elements. In a permutation, the order matters, while in a combination, the order does not matter. In other words, with permutations, the arrangement of items is counted, whereas with combinations, only the selection of items is counted.