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five oranges and five apples have to be placed in 10 slots such that no orange is next to any orange and no apple is next to any apple, in how many different ways can the oranges and apples be placed?

MBA.Aspirant · Answer

winniethepooh wrote:five oranges and five apples have to be placed in 10 slots such that no orange is next to any orange and no apple is next to any apple, in how many different ways can the oranges and apples be placed?

I think there're 2 scenarios: either we put apple, orange, apple, orange..etc or the other way around orange, apple, orange, apple..etc

Slot 1 can be decided in 10 ways (say 5 apples / 5 oranges)
Slot 2 10 ways (5 oranges / 5 apples)
3 8 ways (4 apples/4 oranges )
4 8 ways (4 oranges/ apples)
5 6 ways (3 apples/ 3 oranges)
6 6
7 4
8 4
9 2
10 2

Total = 10*10*8*8*6*6*4*4*2*2 = 14745600 ways

Is this the OA?

winniethepooh · Answer

no

winniethepooh · Answer

I mean the answer is write but ur calculation is wrong! 10*10*8*8*6*6*4*4*2*2=14745600 !! Which is too much!

MBA.Aspirant · Answer

edited. Is this the OA?

Ian Stewart · Answer

First it's not a clearly worded question. If the apples are all identical and the oranges are all identical, the answer is just 2 - the ordering either starts with an apple or starts with an orange, and the rest of the slots are then determined.

If the apples and oranges are all different, we have 10 choices for how to fill the first slot. We then have 5 choices for the next slot (it has to be the other type of fruit). The third slot must be the same type as the first, so we have 4 choices there, and the same happens for the fourth slot. From there the choices fall by 1 each time:

10*5*4*4*3*3*2*2*1*1

You could write that in a few ways - it's just equal to (2)(5!)(5!) for example.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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MBA.Aspirant · Answer

Ian Stewart wrote:First it's not a clearly worded question. If the apples are all identical and the oranges are all identical, the answer is just 2 - the ordering either starts with an apple or starts with an orange, and the rest of the slots are then determined.

If the apples and oranges are all different, we have 10 choices for how to fill the first slot. We then have 5 choices for the next slot (it has to be the other type of fruit). The third slot must be the same type as the first, so we have 4 choices there, and the same happens for the fourth slot. From there the choices fall by 1 each time:

10*5*4*4*3*3*2*2*1*1

You could write that in a few ways - it's just equal to (2)(5!)(5!) for example.

My question is for the 2nd slot, you can either choose from 5 apples or 5 oranges (depending on which you started with in the 1st slot)...so shouldn't this make it 10 ways of choosing taking in consideration the 2 scenarios?

winniethepooh · Answer

yes the answer is 28800!

winniethepooh · Answer

According to me...for the first seat(either apple or orange) there are 10 slots.Say orange takes the slot.
Then for the second orange, the available positions are 3rd,5th,7th,9th = 4 places
Similarly, for the third orange, there are three available places.For the 4th orange there are 2 available places and for the last orange there is a single place.
=10 * 4 * 3 * 2 * 1
= 240 ways

Now, for the apples:
For the 1st apple, there are 2nd,4th,6th,8th and 10th slots = 5 places.
For the 2nd apple = 4 places
For the 3rd apple = 3 places
For the 4th apple = 2 places
For the 5th apple = 1 place.
= 5 * 4 * 3 * 2 * 1 = 120 ways

There for total = 240 * 120 = 28800 ways.

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