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How many positive 4-digit integers contain the digit grouping 73 at least once? For instance, 2738 and 7344 are two such integers to include, but 2378 and 5703 do not meet the restrictions. 

 Jul 3, 2021
 #1
avatar+41 
+1

cases: 73 is in the front, middle or end. 

 case where it is the front: 

first - second digit, 1 choice each

3rd digit: 1~9 = 9

4th : 9

case where it is the middle:

first digit has 9 choices, and the 4th also has 9 choices. 2nd and 3rd have one choice. 

 

case where it is the end. 

same thing, first digit 9 choices, 2nd digit 9 choices, 3 and 4, 1 choice. 

 

So! 9 times 9+9 times 9+9 times 9 = ill leave that to you

 Jul 3, 2021
 #2
avatar+34328 
+2

73 xx       x can be 0-9        100 choices

x73y        x can be 1- 9   y can be 0 - 9    90 choices

xy 73       x can be 1-9    y can be 0-9      90 choices

 

total choices = 280

 Jul 3, 2021
 #3
avatar+41 
+1

oh sorry i didnt see that the first case could have zeros *smh*

 Jul 3, 2021
 #4
avatar+258 
+1

cases: 73 is in the front, middle or end. 

 case where it is the front: 

first - second digit, 1 choice each

3rd digit: 1~9 = 9

4th : 9

case where it is the middle:

first digit has 9 choices, and the 4th also has 9 choices. 2nd and 3rd have one choice. 

case where it is the end. 

same thing, first digit 9 choices, 2nd digit 9 choices, 3 and 4, 1 choice. 

So! (9 times 9)+(9 times 9)+(9 times 9) = ill leave that to you

 Jul 4, 2021

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