Difference between revisions of "2021 AMC 12A Problems/Problem 15"
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==Solution 1 (Bijection)== | ==Solution 1 (Bijection)== | ||
− | <b> | + | Let <math>t</math> be the number of tenors selected and <math>b</math> be the number of basses selected. The requirements are <math>t\equiv b\pmod{4}</math> and <math>(t,b)\neq(0,0).</math> |
+ | |||
+ | It follows that <math>b'=8-b</math> is the number of basses not selected. Since the ordered pairs <math>(t,b)</math> and the ordered pairs <math>(t,b')</math> have one-to-one correspondence, we consider the ordered pairs <math>(t,b')</math> instead. The requirements become <math>t\equiv 8-b'\pmod{4}</math> and <math>(t,8-b')\neq(0,0),</math> which simplify to <math>t+b'\equiv8\pmod{4}</math> and <math>(t,b')\neq(0,8),</math> respectively. | ||
==Solution 2 (Generating Functions)== | ==Solution 2 (Generating Functions)== | ||
− | The problem can be done using a roots of unity filter. Let <math>f(x,y)=(1+x)^8(1+y)^6</math>. By expanding the binomials and distributing, <math>f(x,y)</math> is the generating function for different groups of basses and tenors. That is, | + | The problem can be done using a roots of unity filter. Let <math>f(x,y)=(1+x)^8(1+y)^6</math>. By expanding the binomials and distributing, <math>f(x,y)</math> is the generating function for different groups of basses and tenors. That is, <cmath>f(x,y)=\sum_{m=0}^8\sum_{n=0}^6 a_{mn}x^my^n,</cmath> |
− | <cmath>f(x,y)=\sum_{m=0}^8\sum_{n=0}^6 a_{mn}x^my^n,</cmath> | + | where <math>a_{mn}</math> is the number of groups of <math>m</math> basses and <math>n</math> tenors. What we want to do is sum up all values of <math>a_{mn}</math> for which <math>4\mid m-n</math> except for <math>a_{00}=1</math>. To do this, define a new function <cmath>g(x)=f(x,x^{-1})=\sum_{m=0}^8\sum_{n=0}^6 a_{mn}x^{m-n}=(1+x)^8(1+x^{-1})^6.</cmath> |
− | where <math>a_{mn}</math> is the number of groups of <math>m</math> basses and <math>n</math> tenors. What we want to do is sum up all values of <math>a_{mn}</math> for which <math>4\mid m-n</math> except for <math>a_{00}=1</math>. To do this, define a new function | + | Now we just need to sum all coefficients of <math>g(x)</math> for which <math>4\mid m-n</math>. Consider a monomial <math>h(x)=x^k</math>. If <math>4\mid k</math>, then <cmath>h(i)+h(-1)+h(-i)+h(1)=1+1+1+1=4.</cmath> |
− | <cmath>g(x)=f(x,x^{-1})=\sum_{m=0}^8\sum_{n=0}^6 a_{mn}x^{m-n}=(1+x)^8(1+x^{-1})^6.</cmath> | ||
− | Now we just need to sum all coefficients of <math>g(x)</math> for which <math>4\mid m-n</math>. Consider a monomial <math>h(x)=x^k</math>. If <math>4\mid k</math>, | ||
− | <cmath>h(i)+h(-1)+h(-i)+h(1)=1+1+1+1=4.</cmath> | ||
Otherwise, | Otherwise, | ||
<cmath>h(i)+h(-1)+h(-i)+h(1)=0.</cmath> | <cmath>h(i)+h(-1)+h(-i)+h(1)=0.</cmath> |
Revision as of 11:35, 24 August 2021
Contents
Problem
A choir director must select a group of singers from among his tenors and
basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of
, and the group must have at least one singer. Let
be the number of different groups that could be selected. What is the remainder when
is divided by
?
Solution 1 (Bijection)
Let be the number of tenors selected and
be the number of basses selected. The requirements are
and
It follows that is the number of basses not selected. Since the ordered pairs
and the ordered pairs
have one-to-one correspondence, we consider the ordered pairs
instead. The requirements become
and
which simplify to
and
respectively.
Solution 2 (Generating Functions)
The problem can be done using a roots of unity filter. Let . By expanding the binomials and distributing,
is the generating function for different groups of basses and tenors. That is,
where
is the number of groups of
basses and
tenors. What we want to do is sum up all values of
for which
except for
. To do this, define a new function
Now we just need to sum all coefficients of
for which
. Consider a monomial
. If
, then
Otherwise,
is a sum of these monomials so this gives us a method to determine the sum we're looking for:
(since
and it can be checked that
). Hence, the answer is
.
~lawliet163
Solution 3 (Casework and Vandermonde's Identity)
By casework, we construct the following table. In the last column, we rewrite some of the combinations using the identity
We apply Vandermonde's Identity to find the total number of such groups:
~MRENTHUSIASM
Solution 4 (Enumeration)
By casework, we construct the following table:
We find the total number of such groups:
~sugar_rush
~MRENTHUSIASM
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=FD9BE7hpRvg&t=533s
Video Solution by Hawk Math
https://www.youtube.com/watch?v=AjQARBvdZ20
Video Solution by OmegaLearn (Using Vandermonde's Identity)
https://www.youtube.com/watch?v=mki7xtZLk1I
~pi_is_3.14
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.