F.6 maths(5)

2010-03-31 7:06 am

The letters of the word 'MATHEMATICS' are arranged in a row.
How many arrangements can be made if

a) there is no restriction?
b) the two M's are NOT together?
c) the arrangement begins with the letter A and end with letter A?

回答 (1)

用戶2053

2010-03-31 7:42 am

✔ 最佳答案

Total 11 letters.
Repeated numbers : 2 of M , 2 of A , 2 of T ,
a)
11! / [(2!)(2!)(2!)]
= 4989600 arrangements can be made
b)
Method 1 :
All arrangements - the ways of two M's are together (MM as a letter)
4989600 - 10! / (2!)(2!) = 4082400 arrangements can be made.
Method 2 :
Consider the ways of ATHE ATICS , then insert 2 M to it.
The ways of ATHE ATICS : 9! / (2!)(2!) = 90720
The ways of insert 2 M to it (10 positions) :
90720 x 10C2 = 4082400 arrangements can be made.
c)
= The ways of M THEM TICS : 9! / (2!)(2!) = 90720 arrangements can be made.

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