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Answer to Question #11146 in Statistics and Probability for Ke_091

Question #11146
In how many ways the letters of the word ‘BANGALORE’ can be arranged so that there are no repetitions.
Expert's answer
The word BANGALORE is made up of two 'A's, one 'B', one 'N', one 'G',
one 'L'
and one 'R'.
First consider the two 'A's.
There are a total of 9 positions
that the two 'A's can go. So by the
combination formula, the number of
combinations formed is:
9C2 (pronounced as 9 choose 2). This equals
9!/(2!(9-2)!) = 9!/2!7! =
(9*8*7*6*5*4*3*2*1)/(2*1)(7*6*5*4*3*2*1) = 72/2 =
36.
Now consider the one 'B'.
After the two 'A's have been placed
(wherever they may be), there are
7 positions left that one 'B' can go. By
the combination formula, the
number of combinations formed is:
7C2 =
7!/(2!(7-2)!) = 7!/2!5! = (7*6*5*4*3*2*1)/(2*1)(5*4*3*2*1) = 42/2 = 21.
Now
consider the one 'N'.
After the two 'A's and the one 'B' have been placed,
there are only 6
positions left that the one 'N' can go. By the combination
formula,
the number of combinations formed is:
6C2 = 6!/(2!(6-2)!) =
6!/2!4! = (6*5*4*3*2*1)/(2*1)(4*3*2*1) = 30/2 = 15.
Now consider the one
'G'.
After the two 'A's, the one 'B' and the one 'N' have been
placed,
there is only 5 more position left that the one 'G' can go. By
the
combination formula, the number of combinations formed is:
5C2 =
5!/(2!(5-2)!) = 5!/2!3! = (5*4*3*2*1)/(2*1)(3*2*1) = 20/2 = 10.
Now consider
the one 'L'.
After the two 'A's, the one 'B', the one 'N' and the one 'G'
have been
placed, there is only 4 more position left that the one 'L' can go.
By
the combination formula, the number of combinations formed is:
4C2 =
4!/(2!(4-2)!) = 4!/2!2! = (4*3*2*1)/(2*1)(2*1) = 12/2 = 6.
Lastly, consider
the one 'R'.
After the two 'A's, the one 'B', the one 'N', the one 'G' and
the one
'L' have been placed, there is only 3 more position left that the
one
'R' can go. By the combination formula, the number of
combinations
formed is:
3C2 = 3!/(2!(3-2)!) = 3!/2!1! = (3*2*1)/(2*1)(1) =
3.

Hence, by the multiplication principle, the number of ways the
word
BANGALORE can be arranged if all the letters are used
is:
36*21*15*10*6*3 = 2041200.

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