Since you say "combinations", order does not matter.
However I still can't tell which of two possibilities you want.
The correct solution is either choice (a) or choice (d).
If you mean selecting 4 DIFFERENT letters, there
are only 8 different letters in COMBINATION, which
are {A,B,C,I,M,N,O,T}, and that is
8C4 = 70 choice (a)
Here they all are, 7 rows of 10 each in alphabetical
order:
ABCI ABCM ABCN ABCO ABCT ABIM ABIN ABIO ABIT ABMN
ABMO ABMT ABNO ABNT ABOT ACIM ACIN ACIO ACIT ACMN
ACMO ACMT ACNO ACNT ACOT AIMN AIMO AIMT AINO AINT
AIOT AMNO AMNT AMOT ANOT BCIM BCIN BCIO BCIT BCMN
BCMO BCMT BCNO BCNT BCOT BIMN BIMO BIMT BINO BINT
BIOT BMNO BMNT BMOT BNOT CIMN CIMO CIMT CINO CINT
CIOT CMNO CMNT CMOT CNOT IMNO IMNT IMOT INOT MNOT
--------------------------------------------------
However if you allow such as AIIN or NNOO, where
we can choose two Is, two Ns, or two O's, then the
answer is 136, choice (d).
Here's how to work that:
Case 1: All four are different letters:
8C4 = 70 (Same as the problem as interpreted above)
Case 2: There is exactly one pair of like letters.
Choose the letter of which there will be two of from {I,N,O}
That's 3C1 = 3 ways.
That leaves 7 letters to choose for the remaining two letters:
That's 7C2 = 21
So there are 3*21 = 63
Case 3: There are two pairs of like letters.
Choose 2 from {I,N,O}
That's 3C2 = 3 ways
Total from the 3 cases: 70+63+3 = 136, choice (d)
Here they all are. 13 rows of 10 each and the bottom
row has only 6, in alphabetical order:
ABCI ABCM ABCN ABCO ABCT ABII ABIM ABIN ABIO ABIT
ABMN ABMO ABMT ABNN ABNO ABNT ABOO ABOT ACII ACIM
ACIN ACIO ACIT ACMN ACMO ACMT ACNN ACNO ACNT ACOO
ACOT AIIM AIIN AIIO AIIT AIMN AIMO AIMT AINN AINO
AINT AIOO AIOT AMNN AMNO AMNT AMOO AMOT ANNO ANNT
ANOO ANOT AOOT BCII BCIM BCIN BCIO BCIT BCMN BCMO
BCMT BCNN BCNO BCNT BCOO BCOT BIIM BIIN BIIO BIIT
BIMN BIMO BIMT BINN BINO BINT BIOO BIOT BMNN BMNO
BMNT BMOO BMOT BNNO BNNT BNOO BNOT BOOT CIIM CIIN
CIIO CIIT CIMN CIMO CIMT CINN CINO CINT CIOO CIOT
CMNN CMNO CMNT CMOO CMOT CNNO CNNT CNOO CNOT COOT
IIMN IIMO IIMT IINN IINO IINT IIOO IIOT IMNN IMNO
IMNT IMOO IMOT INNO INNT INOO INOT IOOT MNNO MNNT
MNOO MNOT MOOT NNOO NNOT NOOT
You'll have to ask your teacher which answer
he/she wants.
Edwin
