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how many ways can the committee be formed if Mr. Rudd (teacher) will not serve unless prof. Thudd (academic) is also selected.

I tried this multiple ways, firstly I added the ways of getting no Mr. Thudd and no prof. Rudd with the ways of getting prof. Rudd and all the other teachers including Thudd $9C2$ x $34C4+9C1$ x $35C4 =2140776$

I then tried adding the number of ways of getting Rudd and Thudd with the number of ways to have Rudd out and got the same answer. I also tried it one more method and got the same answer. However this answer is wrong and I can't see what I've done wrong as I believe I've prevented any overcounting. Any help would be much appreciated

Paul
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2 Answers2

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There are: $$\binom{35}4\times\binom{10}2$$ ways if we do not pay attention to the constraint mentioned in your question.

Now we subtract the number of ways where Rudd is selected and Thud is not selected to get the final result:$$\binom{35}4\times\binom{10}2-\binom{34}3\times\binom92$$

drhab
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If no condition, it should be $35C4 \times 10C2$ number of ways.

With the conditions, we have three cases

  1. Mr. Rudd & Prof. Thudd are selected: $34C3 \times 9C1$
  2. Mr. Rudd & Prof. Thudd are not selected: $34C4 \times 9C2$
  3. Mr. Rudd is not selected while Prof. Thudd is: $34C3 \times 9C1$

Sum them and you will get the number of ways to form a committee.

Eddy Piedad
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    this doesn't include the case of Prof. Thudd and no Mr. Rudd – Paul May 05 '22 at 13:05
  • Edited. Thanks. – Eddy Piedad May 05 '22 at 13:08
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    Your count for when Mr. Rudd and Professor Thudd are selected is not correct. If Mr. Rudd is selected, then the remaining three teachers must be selected from among the remaining $34$ teachers. Similarly, if Professor Thudd is selected, then the remaining academic must be selected from among the remaining $9$ academics. Also, you should check your count for the case in which Mr. Rudd is not selected and Professor Thudd is selected. – N. F. Taussig May 05 '22 at 13:10
  • That's right. I edited already. – Eddy Piedad May 05 '22 at 13:12
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    for case 3 you have it should be 9C1 not 9C2 as you have already selected Prof. Thudd and as a result will have 9 academics remaining for one selection – Paul May 05 '22 at 13:19
  • Thanks @Paul. Just edited. – Eddy Piedad May 05 '22 at 13:26