SOLUTION: What is the number of different five-digit numbers that can be formed from the set S={2,3,4,5,6} such that one digit is repeated twice and another digit is repeated twice

           Solve step by step



(a)  Let's consider more simple problem first:

         Given three different symbols A, B and C as an alphabet, 
         how many different 5-symbol words can be formed
         such that symbol A is used twice, symbol B is used twice 
         and symbol C is used once?


         For this problem, we can select a position for A in   = 10 ways;
                                         a position for B in   = 3 ways;
                                         then C occupies the remaining 5th position, at no choice.

         So, for this problems, there are 10*3 = 30 different words.



(b)  Now, returning to the given problem, we can assign any of 5 possible digit to A;
                                                        any of remaining 4 digits to B,
                                                    and any of remaining 3 digits to C.
  

     It gives 5*4*3 = 60 different choices for assigning; but for us, the pairs (A,B) and (B,A)
     are mutually-interchangeable : they lead to the same numbers.


     Therefore, this number 60 we should divide by 2 and take 60/2 = 30.



(c)  Now the final answer to the problem is  30 from (a) multiplied by 30 from (b)

     giving the  ANSWER  30 * 30 = 900.

When a digit is repeated three times, it is repeated two times. 

They are either the arrangements of XXYYZ or of XXYYY, where different
letters represent different digits.

Case 1.  The arrangements of XXYYZ.
Choose the X 5 ways.
Choose the Y 4 ways.
Choose the Z 3 ways.
That's (5)(4)(3)=60 ways.

We divide by two because, for instance, XXYYZ could be chosen respectively, as
33556 or 55336, which are the same.

So that's 30 ways. to select the three digits.

They may be ordered in  distinguishable ways.

That's (30)(30) = 900 ways

Case 2: The arrangements of XXYYY

Choose the digit for X in 5 ways.
Choose the digit for Y in 4 ways.
 
So that's (5)(4) = 20 ways.

Each may be permuted in  distinguishable ways.

That's (20)(10) = 200

For both cases, that's 900+200 = 1100.

Edwin

Sorry, but if you do something three times, you've done it twice.

If 6 digit numbers had been asked for, 333555 would have been acceptable.

Edwin

All the answers above, including mine, are wrong.

"To repeat" means "to do it a second time".

If a digit is repeated, then it is used TWICE.

If a digit is repeated TWICE, then it is used THREE TIMES.

The problem states: 

"one digit is repeated twice and another digit is repeated twice"

This requires a number with at least six digits.

Since the question asks "What is the number of five-digit numbers...",

the answer is clearly 0.

Edwin

For Ikleyn's information concerning the definition of the word "repeat",
here is the definition, copied and pasted below from an online dictionary:

repeat                                   <--the word under discussion
verb [ T ]                               <--the part of speech (verb transitive) 
/rɪˈpit/                                 <--the pronunciation
to say, tell, or do something again:     <--the definition
She repeated the question.               <--an example of usage   
Listen to the tape and repeat each word. <--another example of usage

Ikleyn makes the mistake of thinking that "to repeat TWICE" means the same
thing as "to repeat ONCE". 

Edwin