M203 20260502 Counting and Probability
#Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function "Function (mathematics)") between two sets "Set (mathematics)") such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Given a function f: A-->B, the image of an element a∈A is the element f(a)∈B in the codomain. The pre-image of an element b∈B is any element a∈A in the domain such that f(a)=b. Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.
A function is bijective if and only if it is invertible; that is, a function f:X→Y![media/41d0e184c9edbefc4e796ad04ea93967_MD5.svg]] is bijective if and only if there is a function g:Y→X,
the inverse of f, such that each of the two ways for [composing the two functions produces an identity function: g(f(x))=x
for each x
in X
and f(g(y))=y
for each y
in Y.
2010 AMC 10A Problems/Problem 22
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
$\textbf{(A)}\ 28 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 70 \qquad \textbf{(D)}\ 84 \qquad \textbf{(E)}\ 140$
2012 AMC 10B Problems/Problem 22
Let ($a_1$, $a_2$, ... $a_{10}$) be a list of the first 10 positive integers such that for each $2\le$ $i$ $\le10$ either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?
$\textbf{(A)}\ \ 120\qquad\textbf{(B)}\ 512\qquad\textbf{(C)}\ \ 1024\qquad\textbf{(D)}\ 181,440\qquad\textbf{(E)}\ \ 362,880$
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 4 (Recursion))
- 6 Solution 5
- 7 Solution 6 (similar to solution 2, explained differently))
2010 AMC 10B Problems/Problem 22
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
$\textbf{(A)}\ 1930 \qquad \textbf{(B)}\ 1931 \qquad \textbf{(C)}\ 1932 \qquad \textbf{(D)}\ 1933 \qquad \textbf{(E)}\ 1934$
2017 AMC 10A Problems/Problem 23
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$
2010 AMC 10B Problems/Problem 23
The entries in a $3 \times 3$ array include all the digits from $1$ through $9$, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?
$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 60$
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3 (Doesn't require the hook length formula, copied from Solution 1 and the Beauty of Math video))
- 5 Video Solution by Pi Academy (Fast and Easy))
- 6 Video Solution 2: TheBeautyOfMath
2011 AMC 10A Problems/Problem 22
Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
$\textbf{(A)}\ 2520\qquad\textbf{(B)}\ 2880\qquad\textbf{(C)}\ 3120\qquad\textbf{(D)}\ 3250\qquad\textbf{(E)}\ 3750$