nachoalejandre wrote:
Hi! I really cannot understand why my approach is incorrect.
First, I see how many groups of 3, 2 and 1 I can make; each number of groups I multiply by the number of possible arrangements within each group.
1BALL:
Groups: 26!/(1!25!)=26
Arrangements: 1!=1
Product: 26x1=26
2BALLS:
Groups: 26!/(2!24!)=325
Arrangements: 2!=2
Product: 26x1=650
3BALLS:
Groups: 26!/(3!23!)=2600
Arrangements: 3!=6
Product: 26x1=15600
Sum of the products: 26+650+15600=16276
Where I am reasoning wrong?
Thanks in advance!
Hi
nachoalejandre,
I see where you have made a couple of mistakes, let me help you out.
You have split the problem into three cases, let us approach it the same way.
Case (i) Groups: 26C1 = 26!/(1!25!) = 26
Number of arrangements = 1!
Therefore, the total number of ways = 26*1 = 26
This is correct
Case (ii)Groups: 26C2 = 26!/(2!24!) = 325
Number of arrangements = 2!
Therefore, the total number of ways = 325*2 = 650
Note that although your calculation is correct, you have missed out on the following statement in the question stem, "
If the letters may be repeated". Hence, in the above calculation, entries such as AA, BB, CC, ...ZZ are not accounted for. Accounting for these entries will give you additional 26 entries that have been omitted in your approach.
Therefore, the total number of ways will be 650+26 = 676
Case (iii)Groups: 26C3 = 26!/(3!23!) = 2600
Number of arrangements = 3!
Therefore, the total number of ways = 2600*6 = 15600
Again, your calculations are correct, however, this time again, the same statement that you had overlooked has come back to bite you in two different ways:
a. Three repeating entries such as AAA, BBB, CCC, ..., ZZZ that you haven't accounted for, which will again result in 26 additional entries.
b. Two repeating and one unique entries such as AAB, AAC, ... ZZY, which will result in additional 1950 (26*25*3) entries that have been omitted in your approach.
Therefore, the total number of ways will be 15600+1950+26 = 17,576
Sum total = Case (i) + Case (ii) + Case (iii) = 26+676+17,576 = 18,278
Alternatively, if you had not overlooked that statement, you may have considered: 26+(26*26)+(26*26*26) = 18,278