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# PSL4#63

+1
267
6 Answer: 1/253

Aug 3, 2019

#4
+2

Here's another approach: what are the odds that the group ronnie is in contains ben and javon? there are $$\binom{23}{2}$$ combinations for possible partners for ronnie, one of those options is {ben, javon}, so the odds are $$\frac{1}{\binom{23}{2}}=\frac{1}{253}$$

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Aug 3, 2019

#1
+2

there are 24C3 ways to choose 3 students from 24

then

21C3 ways to choose 3 from the remaining 21

continuing with the pattern

there are

24C3 * 21C3 * ....... * 3C3   ways to split 24 kids into 8 ordered groups of 3 BUT our groups are not ordered so I have to divide by 8!

similarly

if we put Ronnie, Ben  and Jevon into 1 group then we are left with 21 kids to put into 7 groups. So we get

21C3 * 18C3 * ....... 3C3     then we must divide by 7!

To the probability that those 3 boys will end up in the same group will be

$$=\frac{21C3 * 18C3 * ....... 3C3 }{7!}\div \frac{24C3 * 21C3 * 18C3 * ....... 3C3 }{8!}\\ =\frac{21C3 * 18C3 * ....... 3C3 }{7!}\times \frac{8!}{24C3 * 21C3 * 18C3 * ....... 3C3 }\\ =\frac{1 }{1}\times \frac{8}{24C3 }\\ =8\div \frac{24!}{3!21!}\\ =8\times \frac{3!21!}{24!}\\ =8\times \frac{6}{22*23*24}\\ =\frac{6}{22*23*3}\\ =\frac{1}{11*23}\\ =\frac{1}{253}\\$$

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Aug 3, 2019
#2
+2

Melody: I'm no good at this stuff!, but what do think of this naive way:
There are 24C3 =2024 ways of splitting 24 kids.
There are 24/3 =8 ways of splitting them into groups of 3
The probability that Ronnie, Ben and Jevon are in the same group of 3's is: 8 / 2024 = 1 /253

Aug 3, 2019
#3
+1

Good question :)

Well if I word it differently I might be able to make sense of those calculations

There are 24C3 ways to choose  one group of 3 kids out of 24 kids.

So the chance that our boys will be in any one group is 1/24C3

But there are a total of 8 groups to chose from and our boys can be any of those triads.

so that is a probability of 8/ 24C3

I am not completely comfortable with this but the numbers do work and that does seem more than conincidental....

So maybe .....   not sure though.

Melody  Aug 3, 2019
edited by Melody  Aug 3, 2019
#4
+2

Here's another approach: what are the odds that the group ronnie is in contains ben and javon? there are $$\binom{23}{2}$$ combinations for possible partners for ronnie, one of those options is {ben, javon}, so the odds are $$\frac{1}{\binom{23}{2}}=\frac{1}{253}$$

Guest Aug 3, 2019
#5
+1

Yes I think i like that one better.  that makes good sense I think....

Melody  Aug 3, 2019
#6
+1

Thanks for all your help guys!!!!

dgfgrafgdfge111  Aug 3, 2019