The number of different strings that can be made from the letters in ORONO using some or all of the letters is 63
<h3>What is the number of permutations in which n things can be arranged such that some groups are identical?</h3>
Suppose there are n items.
Suppose we have
sized groups of identical items.
Then the permutations of their arrangements is given as
We're given the word ORONO.
- Case 1: One letter string:
3 ways: O, R, or N
- Case 2: Two letter string:
Subcase: Consists O:
5 ways: OR, RO, ON, NO, OO
Subcase: Doesn't consist O:
2 way: RN, NR
Total permutation under this case = 5+2 = 7
- Case 3: Three letter string:
Subcase: Consists 3 'O':
1 way: OOO
Subcase: Consists 2 'O':
6 ways: OOR, ROO, ORO, OON, NOO, ONO
Subcase: Consists 1 'O':
6 ways: ORN, ONR, RNO, RON, NOR, NRO (or that there are 3 distinct letters to be arranged, which can be done in 3! = 6 ways)
Subcase: Doesn't consist O:
0 ways as only R and N cannot form three letter string.
Total permutation under this case = 1+6+6= 13
Subcase: Consists 3 'O':
So, we've got {O,O,O,R,N}
3 'O's are mandatory, so fourth letter is either R or N:
O, O, O, R can arrange themselves in : (as total 4 words but 3 are identical)
Similarly, O, O, O, N can arrange themselves in 4 ways,
So when 4 letter string from ORONO consists of 3 'O's, then total 4+4=8 distinct strings can be made.
Subcase: Consists 2 'O':
So, we've got {O,O,R,N}
The total permuations of these 4 letters, of which 2 are identital is:
Subcase: Consists 1 'O':
0 ways: Single 'O' and one-one R and N cannot form four letter string.
Subcase: Doesn't consist O:
0 ways as only R and N cannot form four letter string.
Total permutation under this case = 8+12 = 20
- Case 5: Five lettered string:
All 5 letters of ORONO would be used.
There are 3 identical objects, so total number of their permutation is:
5!/3! = 20
Thus, from all these cases, we conclude that:
Total permutations of string letters using some or all letters of ORONO is:
3+7+13+20+20 = 63
Thus, the number of different strings that can be made from the letters in ORONO using some or all of the letters is 63
Learn more about permutations here:
brainly.com/question/13443004
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