In the word "examination",
there are two each of letters 'a', 'i', and 'n', which are indistinguishable,
and there is one each of letters 'e', 'o', 'm', 't', and 'x'.
Case 1. Those permutations which contain a maximum of 1 letter from
{a,e,i,m,n,o,t,x} which is P(8,5). That's P(8,5) permutations.
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The following three cases 2,3,and 4 yield the same number of permutations
each:
Case 2. Those which contain two a's, and a maximum of one 'i' or 'n'
We choose the positions for the two a's C(5,2) ways and the remaining 3
from {e,i,m,n,o,t,x} in P(7,3) ways. That's C(5,2)P(7,3) permutations.
Case 3. Those which contain two i's, and a maximum of one 'a' or 'n'
We choose the positions for the two i's C(5,2) ways and the remaining 3
from {a,e,m,n,o,t,x} in P(7,3) ways. That's C(5,2)P(7,3) permutations.
Case 4. Those which contain two n's, and a maximum of one 'a' or 'i'
We choose the positions for the two n's C(5,2) ways and the remaining 3
from {a,e,i,m,o,t,x} in P(7,3) ways. That's C(5,2)P(7,3) permutations.
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The following three cases 5,6, and 7 yield the same number of permutations
each:
Case 5. Those which contain two a's and two i's, and a maximum of one 'n'
We choose the positions for the two a's C(5,2), and the positions for the
two i's C(3,2), and one letter from {e,m,n,o,t,x} in P(6,1) ways. That's
C(5,2)C(3,2)P(6,1) ways.
Case 6. Those which contain two a's and two n's, but a maximum of one i
We choose the positions for the two a's C(5,2), and the positions for the
two n's C(3,2), and one letter from {e,i,m,o,t,x} in P(6,1) ways. That's
C(5,2)C(3,2)P(6,1) ways.
Case 7. Those which contain two i's and two n's, but a maximum of one 'a'
We choose the positions for the two i's C(5,2), and the positions for the
two n's C(3,2), and one letter from {a,e,m,o,t,x} in P(6,1) ways. That's
C(5,2)C(3,2)P(6,1) ways.
Total = P(8,5) + 3·C(5,2)P(7,3) + 3·C(5,2)C(3,2)P(6,1) =
6720 + 3·10·210 + 3·10·3·6 = 6720 + 6300 + 540 = 13560.
Answer: 13,560
Edwin
