In the word DISCRETE how many :
1. 8 letter word can be formed
2. 7 letter word can be formed
3. 6 letter word can be formed
4. 5 letter word can be formed
The 2 E's are indistinguishable. So anytime we use both E's we
must divide by 2!.
1. The number of 8 letter words is = 20160
2. The number of 7 letter words must be broken down into two
cases:
Case 1: Only one E is used: 7! = 5040
Case 2: Both E's are used.
We can choose the 5 other letters to use from these
six {D,I,S,C,R,T}. That's C(6,5) = 6 ways.
For each of those 6 ways there are = 2520
ways to arrange them.
So for case 2 we have 6×2520 = 15120
Total = 5040 + 15120 = 20160
3. The number of 6 letter words must also be broken down into two
cases:
Case 1: No more than 1 E is used.
We choose the 6 letters to use from these {D,I,S,C,R,E,T}
That's C(7,6) = 7 ways
For each of those 7 ways there are 6! = 720 ways
to arrange them.
So for case 1, there are 7×720 = 5040 ways.
Case 2: Both E's are used.
We can choose the 4 other letters to use from these
six {D,I,S,C,R,T}. That's C(6,4) = 6 ways.
For each of those 6 ways there are = 360
ways to arrange them.
So for case 2 we have 6×360 = 2160 ways.
Total = 5040+2160 = 7200 ways.
I'll let you do the 5-letter words.
Edwin