Question 968851: A design requires six distinct interchangeable blocks named A,B,C,D,E and F. In how many ways this design can be assembled. If blocks DEF are together.
Answer by Edwin McCravy(20055)   (Show Source):
You can put this solution on YOUR website! A design requires six distinct interchangeable blocks named A,B,C,D,E and F.
In how many ways this design can be assembled. If blocks DEF are together.
There are 3!=6 ways to arrange the DEF block: DEF,DFE,EDF,EFD,FDE,FED
For each of those 6 ways to arrange the DEF block, there are 4 things to arrange,
A,B,C, and the arranged DEF block. That's 4!=24 ways to arrange the 4 things.
Answer 3!*4! = 6*24 = 144 ways. Here is a computer printout of all 144 ways,
12 rows of 12 each, with the DEF-block in red:
ABCDEF ABCDFE ABCEDF ABCEFD ABCFDE ABCFED ABDEFC ABDFEC ABEDFC ABEFDC ABFDEC ABFEDC
ACBDEF ACBDFE ACBEDF ACBEFD ACBFDE ACBFED ACDEFB ACDFEB ACEDFB ACEFDB ACFDEB ACFEDB
ADEFBC ADEFCB ADFEBC ADFECB AEDFBC AEDFCB AEFDBC AEFDCB AFDEBC AFDECB AFEDBC AFEDCB
BACDEF BACDFE BACEDF BACEFD BACFDE BACFED BADEFC BADFEC BAEDFC BAEFDC BAFDEC BAFEDC
BCADEF BCADFE BCAEDF BCAEFD BCAFDE BCAFED BCDEFA BCDFEA BCEDFA BCEFDA BCFDEA BCFEDA
BDEFAC BDEFCA BDFEAC BDFECA BEDFAC BEDFCA BEFDAC BEFDCA BFDEAC BFDECA BFEDAC BFEDCA
CABDEF CABDFE CABEDF CABEFD CABFDE CABFED CADEFB CADFEB CAEDFB CAEFDB CAFDEB CAFEDB
CBADEF CBADFE CBAEDF CBAEFD CBAFDE CBAFED CBDEFA CBDFEA CBEDFA CBEFDA CBFDEA CBFEDA
CDEFAB CDEFBA CDFEAB CDFEBA CEDFAB CEDFBA CEFDAB CEFDBA CFDEAB CFDEBA CFEDAB CFEDBA
DEFABC DEFACB DEFBAC DEFBCA DEFCAB DEFCBA DFEABC DFEACB DFEBAC DFEBCA DFECAB DFECBA
EDFABC EDFACB EDFBAC EDFBCA EDFCAB EDFCBA EFDABC EFDACB EFDBAC EFDBCA EFDCAB EFDCBA
FDEABC FDEACB FDEBAC FDEBCA FDECAB FDECBA FEDABC FEDACB FEDBAC FEDBCA FEDCAB FEDCBA
Edwin
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