# Combinatorics AMC 10A, 2019 Problem 17

## Competency in Focus: Combinatorics

This problem is based on Combinatorics from American Mathematics contest (AMC 10A, 2019). It includes arrangement of  $n$  items out of which  $r$ items are similar.

## Next understand the problem

[/et_pb_text][et_pb_text _builder_version="4.2.2" text_font="Raleway||||||||" text_font_size="20px" text_letter_spacing="1px" text_line_height="1.5em" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]A child builds towers using identically shaped cubes of different color. 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.) (A) 24 (B) 288 (C) 312 (D) 1260 (E) 40320[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="4.0"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.2.2" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.2.2"]American Mathematical Contest 2019, AMC 10A  Problem 17[/et_pb_accordion_item][et_pb_accordion_item title="Key Competency" _builder_version="4.2.2" open="off"]Combinatorics[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.1" open="off"]4/10[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.2.2" open="off"]

[/et_pb_text][et_pb_tabs _builder_version="4.2.2"][et_pb_tab title="HINT 0" _builder_version="4.0.9"]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title="HINT 1" _builder_version="4.2.2"]Arroanging 8 Cubes is same as arranging 9 cubes and removing 1 cube from the arrangment. It will be same as arranging 8 cubes out of 9 on top of one another.[/et_pb_tab][et_pb_tab title="HINT 2" _builder_version="4.2.2"]If we want to arrange the 9 things then the total no of ways are $9!$.[/et_pb_tab][et_pb_tab title="HINT 3" _builder_version="4.2.2"]But 2 cubes are red (or similiar type), 3 Blue and 4 green so we have to divide the total arrangment by $2!$ , $3!$ and $4!$.[/et_pb_tab][/et_pb_tabs][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built="1" fullwidth="on" _builder_version="4.2.2" global_module="50833"][et_pb_fullwidth_header title="AMC - AIME Program" button_one_text="Learn More" button_one_url="https://www.cheenta.com/amc-aime-usamo-math-olympiad-program/" header_image_url="https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png" _builder_version="4.2.2" title_level="h2" background_color="#00457a" custom_button_one="on" button_one_text_color="#44580e" button_one_bg_color="#ffffff" button_one_border_color="#ffffff" button_one_border_radius="5px"]

AMC - AIME - USAMO Boot Camp for brilliant students. Use our exclusive one-on-one plus group class system to prepare for Math Olympiad

[/et_pb_fullwidth_header][/et_pb_section][et_pb_section fb_built="1" fullwidth="on" _builder_version="4.2.2" global_module="50840"][et_pb_fullwidth_post_slider include_categories="879,878,869" show_arrows="off" show_pagination="off" show_meta="off" image_placement="left" _builder_version="4.2.2" custom_button="on" button_text_color="#0c71c3" button_bg_color="#ffffff" custom_margin="20px||20px||false|false" custom_padding="20px||20px||false|false"][/et_pb_fullwidth_post_slider][/et_pb_section]

This site uses Akismet to reduce spam. Learn how your comment data is processed.

### Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.