Expert: Paul Klarreich - 12/17/2010

Question
QUESTION: How many diffrent 6 digit numbers can be formed using the digits of
the number 5667724? How many of them are divisible by 2?

ANSWER: Questioner: Dhananjay
Country: India
Category: Advanced Math
Private: No
Subject: Maths-Permutation
Question: How many diffrent 6 digit numbers can be formed using the digits of
the number 5667724? How many of them are divisible by 2?
........................................................
I suggest you start by listing the possible sets:

5667724 minus the 5, or 667724
5667724 minus a   6, or 567724
5667724 minus a   7....
5667724 minus the 2....
5667724 minus the 4....

Now for each, apply the 'permutations with repeated elements' formula.  Add them up.

As to 'div by 2', take, for example:

5667724 minus the 5, or 667724

Place '7' at the end, leaving <66724> to be 'permuted'.  This will give you the number of numbers that are NOT div by 2.

................

This should get you going.


---------- FOLLOW-UP ----------

QUESTION: When I tried this lets see the result
Number      Digit kept side  Remaining number  Result (ways )
5667724          5          667724          180
5667724          6          567724          360
5667724          6          567724          360
5667724          7          566724          360
5667724          7          566724          360
5667724          2          566774          180
5667724          4          5667724          180
If I add up last column the result is 1980
But the text book answer is 360 .Now is the answer in text book wrong
i am confused please guide also I request you to elaborate more on numbers  divisible by 2. Please help

Answer
---------- FOLLOW-UP ----------

QUESTION: When I tried this lets see the result
Line  Number      Digit deleted    Remaining digits  Result (ways )
1.  5667724          5          667724          180
2.  5667724          6          567724          360
3.  5667724          6          567724          360
4.  5667724          7          566724          360
5.  5667724          7          566724          360
6.  5667724          2          566774          180
7.  5667724          4          566772          180
If I add up last column the result is 1980
But the text book answer is 360 .Now is the answer in text book wrong
i am confused please guide also I request you to elaborate more on numbers  divisible by 2. Please help
......................
Your calculation for each line seems right.  However,you should delete lines 3 and 5.  (Do you see why?)

That leaves:

Line  Number      Digit deleted    Remaining digits  Result (ways )
1.  5667724          5          667724          180
2.  5667724          6          567724          360
4.  5667724          7          566724          360
6.  5667724          2          566774          180
7.  5667724          4          566772          180
         ----------
         1260

Now as to odd numbers, for each of the lines:

Line   Remaining digits  Result (ways )
1.     667724          180
       Put a 7 at the end, and permute 66724: 5!/2!
2.     567724          360
       Put a 7 at the end, and permute 56724: 5!
       Put a 5 at the end, and permute 67724: 5!/2!
4.     566724          360
        continue in the same way.
6.     566774          180
        continue in the same way.
7.     566772          180
        continue in the same way.