Answer to Question #11146 in Statistics and Probability for Ke_091

Question

In how many ways the letters of the word ‘BANGALORE’ can be arranged so that there are no repetitions.

Answer 1

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.