Couple of Counting (how many ways) questions.

Question

1.If I have a group of 10 seats reserved for people, and there are n=>10 total people, how many ways are there to choose who gets the 10 seats?

for ex:If there was a definite number of people lets say 21 people instead of n>=10 , then it would be 21!/10!11!, but since its n=>10 I cannot seem to figure out the formula for all the results for anything greater or equal to 10.

2.If there a set a(A) of a size i and set b(B) of a size j, i=>j=>1, how would I go about solving how many functions are not one-to-one for f: A->B

for ex:I tried different motions but still getting muffled results(wrong/mixed answers)

3.If there are 30 cards, 6 are red and 7 are black, 10 are not black and red.How many cards are red and black ? --Im assuming it would be along the lines of (see below), am I on the right track or completely off ?

(30) (23) (17)

( 7) (6 ) (10)

^tried styling it didnt work image they are in one bracket and over each other (n/k) type of situation.

4.I have I greater and/or equal to 1, to prove that in any set i+1 ranging from integers (1,2,..,2n) to show that there are going to be 2 consecutive integers.

---I know I need to use pigeonhole principle but I cant budge on the question, I tried implementing it but im positive Im doing it wrong.

Again any guidance would greatly be appreciated.

Answer 1

1. In general the number is given by $\binom{n}{10}=\frac{n!}{10!(n-10)!}$.

2. I suggest you count the total number of functions, and subtract the number of one-to-one functions. (There are formulas for both of these numbers.)

3. If, of the $30$ cards, we have that "10 are not black and red" then $30-10$ are "red and black", right? (Is this phrased exactly how the question was asked?)

4. We have a set $S$ of $n+1$ elements from $\{1,2,\ldots,2n\}$. We set up $n$ pigeonholes $\{1,2\}$, $\{3,4\}$ and so on up to $\{2n-1,2n\}$. At least one of the pigeonholes has both elements in $S$.

Answer 2

For 1, you are correct that without knowing the size of the pool to choose from you can't find the answer.

For 2, can you find how many total functions there are from $A$ to $B$? Now subtract the number that are one-to-one. To get that, choose the range as a subset of $B$, then order it to match the order of $A$

For 3, is "not black and red" supposed to be "not(black and red)" Draw a Venn diagram. The number is small-there are only $30$ cards

For 4 it sounds like $i=n$. Please be careful with your variables. Partition the set into pairs $(2k-1,2k)$ How many pairs is that?