Difference between revisions of "2010 AMC 10B Problems/Problem 5"

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==Solution==
 
==Solution==
In this month there are four weeks and three remaining days. As long as the last three days and the first four days have the same number of Mondays and Wednesdays, then it works. The number of days the month can start on is <math>\boxed{\textbf{(B)}\ 3}</math>
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In this month there are four weeks and three remaining days. Any 7 days must have exactly one Monday and one Wednesday, so it works if the last <math>31 - 4\cdot 7 = 3</math> days have the same number of Mondays and Wednesdays. We have three choices: Monday, Tuesday, Wednesday; Thursday, Friday, Saturday; Friday, Saturday, Sunday. The number of days the month can start on are Monday, Thursday, and Friday, for a final answer of <math>\boxed{\textbf{(B)}\ 3}.</math>
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2010|ab=B|num-b=4|num-a=6}}
 
{{AMC10 box|year=2010|ab=B|num-b=4|num-a=6}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 02:14, 24 November 2019

Problem

A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Solution

In this month there are four weeks and three remaining days. Any 7 days must have exactly one Monday and one Wednesday, so it works if the last $31 - 4\cdot 7 = 3$ days have the same number of Mondays and Wednesdays. We have three choices: Monday, Tuesday, Wednesday; Thursday, Friday, Saturday; Friday, Saturday, Sunday. The number of days the month can start on are Monday, Thursday, and Friday, for a final answer of $\boxed{\textbf{(B)}\ 3}.$

See Also

2010 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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 10 Problems and Solutions

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