Quick Question: number of committees, handshakes possible

[Jaskaran]

Suppose in congress there are 54 democrats and 46 republicans
a)How many seven person committees can be formed from either pary?
So I would set it up as:
54C7
and 46C7
So would I then add both the amounts?
b)How many 4 person committees can be formed entirely republican?
46C7 and evaluate right?


2)there are 10 people in a room. Each shakes hands with each other exactly once. How many shakes will take place?
I assume it's 100 or $\displaystyle 10^2$ or 10C2

3)ADVISORY can be arranged 8! times right? Or 40,320

4)Which entry in the nth row of Pascal's Triangle is the coefficient of $\displaystyle x^n$ in the binomial expansion of $\displaystyle (x+n)^n$? This one I have no clue on how to induce. I assume it's xCy

Lastly,

Consider the sequence defined by:
g1=18
gn + 1= (1/3)gn , n>1

So that one means that every term after 18 is being divided by 3 right?
The first four terms would be (if I'm right): 18,6,2,2/3,2/9
and the sum of the first 10 terms is ~26.99 since (18(1-(1/3)^10)/(1-(1/3))

ALl your valuable help is appreciated much.

 

[pka]

Re: Quick Question: number of committees, handshakes possibl

Suppose in congress there are 54 democrats and 46 republicans
a)How many seven person committees can be formed from either party.

If that means seven from either party then add.
\(\displaystyle \L {54 \choose 7} + {46 \choose 7}\)

b)How many 4 person committees can be formed entirely republican?
\(\displaystyle \L {46 \choose 4}\) Note the four!

2)there are 10 people in a room. Each shakes hands with each other exactly once. How many shakes will take place?
\(\displaystyle \L {10 \choose 2}\) that is the number of two-element subsets.
Each such subset represents one handshake.


3) ADVISORY can be arranged 8! times right? Or 40,320 CORRECT