Solve Quick Questions About Tests: The Binomial Theorem and More

[F.B]

I have a test tomorrow and i really need to get these answers.

1. The streets of a city are laid out in a rectangular grid, that is 7 by 4. The 7 squares are the base and the 4 are the height.
A)How many routes begin with 3 north steps in a row?

The book has 8 and i can count 8 ways but i don't know how to get it.

2. The expansion of (3/x - x^3)^8 find
a) The constant term
b)The term containing x^12

For a) i did this:

C(8,r) x (3/x)^8-r x (-x^3)^r
I can't remember how to do it with a 3 as the top number. i can't add the exponents unless they have the same base so how do i do this.

For b is there an easy way to figure out which has x^12 which actually doing the whole expanding.

3. Use the binomial Theorem to determine the expansion of (a + b + c)^3
I have three terms how do i solve this.

 

[Atomos]

1. I assume you mean shortest routes between two points in opposite corners. If you move 3 north first, then you may now ignore the 3 rows you crossed, and a new 1 x 7 grid results. The number of ways from one corner to the other is your answer. Counting is actually an acceptable stratagy for pathway problems, however, here is a more mathematical approach:
Let us call all horizontal movements by 1 sqaure x, and all vertical movments by one sqaure y. In all we must make 7 x's and 1 y.
x, x, x, x, x, x, x, y
Permute that set of characters.
2. a) write out a list of the degrees of the terms in that binomial in the expansion. Look for which set of degrees will make the x in the numerator have the same degree as the x in the denominator. The 3 means nothing until it comes time to calculating the actual coefficient.
b) it is similar to what you do in 'a', except you done want the dgerees to cancel out. You want there to be a degree 12 left on the x.
3. Group the three terms into two groups and cary out expansion.

 

[F.B]

For number 3 there are so many ways you can group them. You can group them in two ways
(a)(b+c)
(a+b)(c) so which way do i use.