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Hi, I have a question to ask regarding subquestion 3. in the picture. I solved it by using ${10 \choose 2}$ since $2$ of the $4$ houses are fixed already, which I thought would leave me with $10$ choices. However, the answer is $126$, which completely puzzles me, because thinking back, I realised that the $10$ houses are of $5$ different designs, which made me wonder if my answer was wrong. Even so, my method would have overcounted, but the answer is way more than mine, which means I have undercounted instead. I simply cannot find out why am I wrong.

Aditya Agarwal
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Lim LS
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1 Answers1

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  • Ways to select the style B house: $\dbinom 2 1 = 2$.
  • Ways to select the style C house: $\dbinom 3 1 = 3$.
  • Ways to select the rest 2 houses: $\dbinom 7 2 = 21$.

Notice that we choose any 2 houses from the style A, style D and style E collections, since we want exactly one house of style B and exactly one house of style C.

So, the number of combinations is: $2 \cdot 3 \cdot 21 = 126$

thanasissdr
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  • Oh, now I realise that they only want one of B and C. Just one more confusion to clear up though. Why must we have ways to select the style of B and C houses, since I thought they are identical? Similarly, why choose from 7 houses and not, from the 3 remaining designs? – Lim LS Apr 19 '15 at 03:32
  • Apparently, houses are not considered to be identical. For example, we can say the houses in style A are $A_1,A_2$. Houses in style B are $B_1,B_2$ etc. Try to write down your sample space. The exercise says that we pick 4 out of 12 houses. That means your sample space consists of 12 elements. Thus, it treats the houses as discrete objects. So, we have to do the same. – thanasissdr Apr 19 '15 at 04:01
  • @LimLS Note that while the houses in the same style may look the same, they can be distinguished by their locations. – N. F. Taussig Apr 19 '15 at 10:31
  • @N.F.Taussig I don't understand. How can the houses be distinguished by their locations when their locations are not constant? – Lim LS Apr 20 '15 at 13:56
  • For any particular arrangement, the houses in the same style can be distinguished by their locations. – N. F. Taussig Apr 20 '15 at 21:26