Determine the number of permutations of the letters of the word SAMPLE if all are taken at a time.
Answers
Answer:
We have two P's, two R's, three O's and other letters T, I, and N have appeared for once
1. Words with four distinct letters.
We have 6 letters all total (I,N,P,R,O and T) so we can arrange this letter in
6
C
4
×4!=
360
ways
2. Words with exactly a letter repeated twice.
We have P, R and O repeating itself. Now one of this three letter can be choose in
3
C
1
=3 ways
The other two distinct letters can be selected in
5
C
2
=10 ways.
Now each combination can be arranged in
2!
4!
=12 ways.
So total number of such words 3×10×12=
360
3. Words with exactly two distinct letters repeated twice
Two letters out of the three repeating letters P, R and O can be selected in
3
C
2
=3 ways.
Now each combination can be arranged in
2!×2!
4!
=6
So, total number of such words =3×6=
18
4. Words with exactly a letter repeated thrice
We have one portion for this as our main letter that is O.
Now we have to select 1 letter out of the 5 remaining options so number of ways to select this
5
C
1
=5 ways
Now each combination can be arranged in
3!
4!
=4
So, total number of such words =5×4=
20
So, all possible number of arrangements =360+360+18+20=758 ways
burning of your brain in this stage is oxidation