## November 29, 2006

### Counting 1..2..3!

At the beginning of today's class, the class was divided into 5 groups of 4. Each group were given the same 5 questions on counting. As seen below. I will now explain briefly how to get these answers and why.Solutions and Explanations

1.a) As seen in question one the answer is 21 terms. Determining how many terms there are is defined by (n +1) , n is the exponent after the binomial expansion. So ( 9+1) =10 and (10 +1)=11. Therefore 10+11 = 21 terms.

1.b) The answer is (x +y) ^10. This is because in order for you to have a middle term, you need to have the same amount of numbers before and after that middle number, and this can only be done with an odd number of terms, but since our formula is (n+1), in finding how many terms, in a binomial expansion it would have to be an even number. (x +y) ^10, has 11 terms therefore having 5 before and 5 after. The middle term would be 6 in this expansion.

c) As said above, If "n" is an even number.

2. The answer is there is no middle term because the "n" was an odd number.

3. a) self explanatory. (18)(10)(20)(10)(20)(10) <given to us,and 10 stands for numbers you can choose between 0-9.

b) KEY WORD IS NEW. So without restrictions it would look like this (26)(10)(26)(10)(26)10) but since how many new ones are formed you would subtract the answer from, part a, from the "no restrictions" to get all the new codes.

4. It is 7P4. It is pick, because order matters, enemies do not want to sit beside enemies, or window sitters. 7 stands for the empty seats the can pick from, and 4 stands for the people who will be sitting in their picked spots.

5. a) 18C6 because order does not matter. 18 stands for number of people. 6 stands for people choosen to be on committee.

b) 8C4(10C2), exactly 4 boys, so since there are 8 boys they have to choose exactly four, and remaining 2 people will have to be chosen from the 10 girls, to make the committee of 6 people.

c) 10C4(8C2)Same explanation as above, but exactly 4 girls have to be chosen from the 10 girls , then the other 2 people will have to be chosen from the 8 boys, to form the committee.