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Four-digit integers are formed using the digits 2, 3, 4 and 5. Any of the digits can be used any number of times. How many such four-digit integers are palindromes? Palindromes read the same forward and backward.

Guest Mar 18, 2015

Best Answer 

 #5
avatar+86890 
+10

After considering this problem again, I believe that my answer  is correct - 16.

Looking at Melody's answers, she states that AABB and BBAA are palindromes.

But after doing a little research, I discovered that this isn't quite correct. A palindromic number or numeral palindrome is a number that remains the same when its digits are reversed. See the description, here....

http://en.wikipedia.org/wiki/Palindromic_number

Notice, if we chose 5 and 2..... 5522 isn't the same as 2255. (They are different numbers)

The only two palindromes we can make here are 2552  and 5225.

To see this more clearly......note that we have 4 ways to choose the first number and 3 ways to choose the second number = 12. And the digits in the 3rd and 4th positions will just be the reverse of this order.

So....these 12 ways plus the 4 ways of choosing the same digit to occupy all four positions gives us only 16 possibilities.

 

  

CPhill  Mar 20, 2015
 #1
avatar+17736 
+5

There are 4 choices for the first digit, 4 choices for the second digit, 4 choices for the third digit, and 4 choices for the last digit; thus, there will by 4 x 4 x 4 x 4 choices.

To determine the number of palindromes: there are 4 choices for the first digit and 4 choices for the second digit, however, there is only 1 choice for the third digit (it must match the second digit), and there is only 1 choice for the last digit (it must match the first digit). Thus, there will be 4 x 4 x 1 x 1 choices.

geno3141  Mar 19, 2015
 #2
avatar+86890 
+5

Any digit "a" could be used 4 times..aaaa..so that's 4

And any two of the digits - "a" and "b" - can be selected out of 4 = 6 ways

And of the second, we have two ways to arrange the digits.... abba, baab

So...4 + 6(2)  = 16 palindromes

 

  

CPhill  Mar 19, 2015
 #3
avatar+92624 
+5

(Mine is the same as CPhill but I think he overlooked a couple of possibilities )

I will need to either choose 2 digits or 1 digit

there are 4C2 = 6 ways of choosing 2 digits and 4 ways of choosing just one digit

Say I chose 2 digits and they are A and B  the palindomes i can make are

AABB

ABBA

BBAA

BAAB          there are only 4

If i choose just one letter, sat A,  then there is only one way to make a palindome, i.e.  AAAA

So

there will be   6*4+4*1 = 24+4 = 28 ways.

Melody  Mar 19, 2015
 #5
avatar+86890 
+10
Best Answer

After considering this problem again, I believe that my answer  is correct - 16.

Looking at Melody's answers, she states that AABB and BBAA are palindromes.

But after doing a little research, I discovered that this isn't quite correct. A palindromic number or numeral palindrome is a number that remains the same when its digits are reversed. See the description, here....

http://en.wikipedia.org/wiki/Palindromic_number

Notice, if we chose 5 and 2..... 5522 isn't the same as 2255. (They are different numbers)

The only two palindromes we can make here are 2552  and 5225.

To see this more clearly......note that we have 4 ways to choose the first number and 3 ways to choose the second number = 12. And the digits in the 3rd and 4th positions will just be the reverse of this order.

So....these 12 ways plus the 4 ways of choosing the same digit to occupy all four positions gives us only 16 possibilities.

 

  

CPhill  Mar 20, 2015
 #6
avatar+92624 
0

Yes of course you are correct Chris.    I must have been suffering a major CDD recurrence.   

(@~_@~)        (@~_@~)         (@~_@~)

Melody  Mar 20, 2015

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