25. Seven students (Catherine, Jim, Jerome, Lucia, Lisa, Melinda, and Dave) are entered in a debate contest.
a) In how many ways can they give their arguments if the following conditions must be met?
i) Lucia must go second: _________________________________________________
ii) Lisa must go first and Dave must go second to last: __________________________
iii) All of the boys go first followed by the girls: _______________________________
iv) All of the girls go first followed by the boys: _______________________________
v) Females and males alternate with a female going first: ________________________
Hi Reiny,
I was solving them as following:
i) Lucia must go second:
_ L _ _ _ _ _ = 6! =6x5x4x3x2x1 = 720
ii) Lisa must go first and Dave must go second to last:
L _ _ _ _ D _ = 5! =5x4x3x2x1 = 120
iii) All of the boys go first followed by the girls:
BBBGGGG = 3!4! =(3x2x1)(4x3x2x1) = (6)(24) = 144
iv) All of the girls go first followed by the boys:
GGGGBBB = 4!3! =(4x3x2x1)(3x2x1) = (24)(6) = 144
v) Females and males alternate with a female going first:
GBGBGBG = 4!3! =(4x3x2x1)(3x2x1) = (24)(6) = 144
Anyone else have any edits as well, I would highly appreciate it.
Thanks for your time.
Fulfill the obligations first
a) XLXXXXX -----> Numbe of ways = 6*1*5*4*3*2*1 = 720
b) LXXXXDX -----> Number of ways = 1*5*4*3*2*1*1 = ...
c) 3 boys, then 4 girls: BBBGGGG -----> 3*2*1*4*3*2*1 = ....
d) 4 girls and 3 boys: GGGGBBB -----> ......... = ....
e) ........
yeah thats what I thought you were doing. I was just talking about question e.) as your answer was ........, I just got confused.
Thankyou so much for your time and effort, I really appreciate it.
To solve these problems, we can use the concept of permutations. A permutation is an arrangement of objects in a specific order.
a) To find the number of ways the students can give their arguments, we can use the concept of permutations. There are 7 students, so there are 7 positions for them to occupy.
i) Lucia must go second:
We fix Lucia in the second position, and the remaining 6 students can occupy the remaining 6 positions. The number of ways to arrange the remaining students is 6!.
Therefore, there are 6! ways for the students to give their arguments if Lucia must go second.
ii) Lisa must go first and Dave must go second to last:
We fix Lisa in the first position, and Dave in the second to last position. The remaining 5 students can occupy the remaining 5 positions. The number of ways to arrange the remaining students is 5!.
Therefore, there are 5! ways for the students to give their arguments if Lisa must go first and Dave must go second to last.
iii) All of the boys go first followed by the girls:
Since there are 7 students with 3 boys and 4 girls, we can fix the order of the boys first. The number of ways to arrange the boys is 3!. Then, we can arrange the girls, which is 4!. Finally, we multiply the number of arrangements for the boys with the number of arrangements for the girls to get the total number of ways.
Therefore, there are 3! * 4! ways for the students to give their arguments if all of the boys go first followed by the girls.
iv) All of the girls go first followed by the boys:
Similarly, we fix the order of the girls first, which is 4!. Then, we fix the order of the boys, which is 3!. Finally, we multiply the number of arrangements for the girls with the number of arrangements for the boys to get the total number of ways.
Therefore, there are 4! * 3! ways for the students to give their arguments if all of the girls go first followed by the boys.
v) Females and males alternate with a female going first:
We fix the order of the females first, which is 4!. Then, we fix the order of the males, which is 3!. Finally, we multiply the number of arrangements for the females with the number of arrangements for the males to get the total number of ways.
Therefore, there are 4! * 3! ways for the students to give their arguments if females and males alternate with a female going first.
Remember to use the factorial function (!) to find the number of ways to arrange a group of objects.