The U.S Olympic gymnastics team for the 2016 Rio Games comprised five women and five men. In how many ways can this group of athletes arrange themselves in a row for a group picture if:
a) There is no restriction
b) The men stand together in a group and the women stand together in a group?
c) The men and women must occupy alternate positions?
a) The 5 women + the 5 men = 10
P(10,10) =10!
b) Women : P(5,5) Men: P(5,5)
p (5,5) * p (5,5) = 120*120=14400
c) p(5,5)*p(5,5)=120*120=14400
Even though your answer is correct,
I would argue c) as follows:
from left to right
MWMWMW...
= 5*5*4*4*3*3*2*2*1*1
= 5! * 5! = 14400
a) Well, there are 10 athletes in total, so they can arrange themselves in a row in 10! (10 factorial) ways. That's a whole lot of ways to take a group picture!
b) If the men stick together and the women stick together, then we can consider the women as one unit and the men as another unit. So, there are 5! (5 factorial) ways to arrange the women and 5! ways to arrange the men. Multiplying these two together gives us a total of 12,400 ways. That's a lot of ways to group them together!
c) If the men and women must alternate positions, then let's think of them as one giant group of 10 athletes. There are 10! ways to arrange this group. However, we have to be careful because there are two "types" of athletes - men and women. So, if we fix the position of one type (let's say women), then there are 5! ways to arrange the women, and similarly, 5! ways to arrange the men in their fixed positions. Multiplying these two together gives us a total of 14,400 ways. So, they can strike some pretty fancy poses!
a) In this case, there is no restriction on how the group of athletes can arrange themselves. We can use the formula for permutation, denoted by P(n,r), where n is the total number of people and r is the number of positions to be filled.
In this case, there are 10 athletes in total, so n = 10. And they need to arrange themselves in a row, which means all 10 positions need to be filled, so r = 10.
Therefore, the total number of ways they can arrange themselves is given by:
P(10,10) = 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800
So, there are 3,628,800 ways the group of athletes can arrange themselves in a row for a group picture without any restrictions.
b) In this case, the men need to stand together in a group and the women need to stand together in a group. This means that we can treat each group as a single entity for the purpose of permutation.
We have 5 women, so we calculate the number of ways they can arrange themselves within their group, which is P(5,5).
We also have 5 men, so we calculate the number of ways they can arrange themselves within their group, which is also P(5,5).
Since the groups can be arranged independently, the total number of ways they can arrange themselves is given by multiplying these two numbers:
P(5,5) x P(5,5) = 120 x 120 = 14,400
So, there are 14,400 ways the group of athletes can arrange themselves in a row for a group picture if the men stand together in a group and the women stand together in a group.
c) In this case, the men and women must occupy alternate positions. This means that we can first calculate the number of ways the men can arrange themselves in alternate positions, which is P(5,5). Then, we calculate the number of ways the women can arrange themselves in alternate positions, which is also P(5,5).
Since the men and women can be arranged independently, the total number of ways they can arrange themselves in alternate positions is given by multiplying these two numbers:
P(5,5) x P(5,5) = 120 x 120 = 14,400
So, there are 14,400 ways the group of athletes can arrange themselves in a row for a group picture if the men and women must occupy alternate positions.
To get the answers to these questions, we can use the concept of permutations.
a) When there are no restrictions, we can arrange the group of athletes in a row using all 10 of them. The number of ways to arrange them is given by the permutation formula P(n, r) = n!, where n is the total number of athletes (10 in this case) and r is the number of positions to fill (also 10 in this case). So, the number of ways to arrange them is 10!.
b) When the men stand together in a group and the women stand together in a group, we can consider the men's group and women's group as two separate entities. Within each group, the individuals can be arranged in a row. So, the number of ways to arrange the women is given by P(5,5), and the number of ways to arrange the men is also P(5,5). To get the total number of ways, we can multiply the number of ways to arrange the women with the number of ways to arrange the men: P(5,5) * P(5,5) = 120 * 120 = 14400.
c) When the men and women must occupy alternate positions, we can consider them as two separate groups. Within each group, the individuals can be arranged in a row as well. Similar to part b, the number of ways to arrange the women is given by P(5,5), and the number of ways to arrange the men is also P(5,5). To get the total number of ways, we can multiply the number of ways to arrange the women with the number of ways to arrange the men: P(5,5) * P(5,5) = 120 * 120 = 14400.
So, the answers are:
a) 10! = 3,628,800 ways
b) 14,400 ways
c) 14,400 ways