Is your family like this?

Probability Level 2

4 brothers are told to stand in a row for their family portrait. Their heights are:

Abe: 4'6''
Ben: 4'
Clark: 3'6''
Dan: 3'

There's also a "Happy Birthday Dad!" poster that has to be held by 2 brothers standing next to each other, and in order to hold it straight, those two brothers have to be at most 6'' apart in height.

Mom's a bit crazy and wants to take a bunch of pictures so she has her sons stand in all 4! possible orders (ABCD, DCBA, DCAB, ... etc.) In how many of the shots can the poster be held evenly?

8 12 22 20 all 24

3 solutions

Zandra Vinegar Staff
Oct 17, 2015

By complementary counting:

In order for the poster to NOT be usable, Ben must be standing on the end, next to only Dan. Similarly, Clark must be on the other end and next to only Abe. That makes BDAC and CADB the only two arrangments in which the poster can't be held evenly. $24 - 2 = \fbox{22}$

NIce problem. If we were to add Edgar at $2' 6''$ to the family and ask a similar question, then I find that $106$ of the $5! = 120$ possible shots would work. If we then added in Francis at $2' 0''$ .....

Brian Charlesworth - 5 years, 8 months ago

Thank you -- both for the complement and for noting this generalization to the problem. :)

Zandra Vinegar Staff - 5 years, 7 months ago

@Zandra Vinegar , I think you wanted to write BDAC and CADB

Mircea Barba - 5 years, 7 months ago

Thanks, you're right. Fixed. :)

Zandra Vinegar Staff - 5 years, 7 months ago

Nicholas Kross
Aug 9, 2018

Note that A is only "within range" of B, and D is only within range of C. B and C are each in range of two other letters:

A---B---C---D

For nobody to be in range of anybody else, B can only connect with D, so B has to be on the end (so it only connect with one other): B---D Same with C and A: A---C These gives us: B---D---A---C and its reversal, C---A---D---B

4! total orderings, minus 2 nobody-in-each-other's-range-orderings, = 4!-2 = 22.

Bostang Palaguna
Aug 18, 2020

In order for the poster not evenly held, every children must stand besides other children whose height differ more than 6''.

for Ben, he can only stands beside Dan, which mean he must be on the edge.

Another person that can stand besides Dan is Abe.

So, there are only 2 'unstable' configuration:

Ben-Dan-Abe-Clark and Clark-Abe-Dan-Ben.

which by using the complementary, there are 22 'stable' configurations.

Is your family like this?

3 solutions

0 pending reports