What is the NO-SHORTCUT approach for learning great Mathematics?

[et_pb_section fb_built="1" _builder_version="4.2.2" custom_padding="||42px|||"][et_pb_row _builder_version="4.2.2" module_alignment="center"][et_pb_column type="4_4" _builder_version="4.2.2"][et_pb_blurb _builder_version="4.2.2"][/et_pb_blurb][et_pb_blurb _builder_version="4.2.2"][/et_pb_blurb][et_pb_team_member name="American Mathematics contest 10 (AMC 10) - Combinatorics problems " _builder_version="4.2.2" header_level="h1" header_font="Acme||||||||" header_text_align="center" header_font_size="28px" body_font="Acme||||||||" body_text_align="center" body_letter_spacing="1px" background_color="#032b35" background_layout="dark" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"] Try these AMC 10 Combinatorics Questions and check your knowledge![/et_pb_team_member][et_pb_image src="https://www.staging18.cheenta.com/wp-content/uploads/2020/01/AMC-8-Geometry-Questions-3.png" _builder_version="4.2.2"][/et_pb_image][et_pb_team_member name="AMC 10A, 2020, Problem 9" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$

$\textbf{(A) } 9 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 27 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 77$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 15" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 23$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 19" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?

$\textbf{(A) } 125 \qquad \textbf{(B) } 250 \qquad \textbf{(C) } 405 \qquad \textbf{(D) } 640 \qquad \textbf{(E) } 810$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 14" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?

$\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 17" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]A child builds towers using identically shaped cubes of different colors. How many different towers with a height $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)

$\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,260 \qquad\textbf{(E) } 40,320$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 20" _builder_version="4.3.1" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?

$\textbf{(A) }\dfrac{1}{21}\qquad\textbf{(B) }\dfrac{1}{14}\qquad\textbf{(C) }\dfrac{5}{63}\qquad\textbf{(D) }\dfrac{2}{21}\qquad\textbf{(E) }\dfrac{1}{7}$

[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2018, Problem 4" _builder_version="4.3.1" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)

$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$

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