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Difference between revisions of "2022 AMC 10A Problems/Problem 24"

(Created page with "==Problem== The numbers <math>3, 5, 7, a,</math> and <math>b</math> have an average (arithmetic mean) of <math>15</math>. What is the average of <math>a</math> and <math>b</ma...")
 
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==Problem==
 
==Problem==
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The numbers <math>3, 5, 7, a,</math> and <math>b</math> have an average (arithmetic mean) of <math>15</math>. What is the average of <math>a</math> and <math>b</math>?
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How many strings of length <math>5</math> formed from the digits <math>0</math>,<math>1</math>,<math>2</math>,<math>3</math>,<math>4</math> are there such that for each <math>j\in\{1,2,3,4\}</math>, at least <math>j</math> of the digits are less than <math>j</math>? (For example, <math>02214</math> satisfies the condition because it contains at least <math>1</math> digit less than <math>1</math>, at least <math>2</math> digits less than <math>2</math>, at least <math>3</math> digits less than <math>3</math>, and at least <math>4</math> digits less than <math>4</math>. The string <math>23404</math> does not satisfy the condition because it does not contain at least <math>2</math> digits less than <math>2</math>.)
  
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<math>\textbf{(A) } 0 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 60</math>
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<math>\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296</math>
  
 
== Solution By Omega Learn with Complementary Counting==  
 
== Solution By Omega Learn with Complementary Counting==  
  
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[youtube]https://www.youtube.com/watch?v=jWoxFT8hRn8&list=PLT9bNzqjDoMl3jNviYrczw7Ck_ArS54Xn&index=8[/youtube]
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https://www.youtube.com/watch?v=jWoxFT8hRn8&list=PLT9bNzqjDoMl3jNviYrczw7Ck_ArS54Xn&index=8

Revision as of 04:07, 12 November 2022

Problem

How many strings of length $5$ formed from the digits $0$,$1$,$2$,$3$,$4$ are there such that for each $j\in\{1,2,3,4\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies the condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, at least $3$ digits less than $3$, and at least $4$ digits less than $4$. The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$.)

$\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296$

Solution By Omega Learn with Complementary Counting

https://www.youtube.com/watch?v=jWoxFT8hRn8&list=PLT9bNzqjDoMl3jNviYrczw7Ck_ArS54Xn&index=8

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