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How many different arrangement can be made by using all the le

Question

How many different arrangement can be made by using all the letters in the MATHEMATICS?
How many of them begin with $C$ ? How many of them begin with $T$ ?

\frac{11 !}{2 ! 2 ! 2 ! ^{'}} \frac{10 !}{2 ! 2 ! 2 ! ^{'}} \frac{10 !}{2 ! 2 !}

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How many different arrangement can be made by using all the letters in the MATHEMATICS?
How many of them begin with

C

? How many of them begin with

T

11 mins ago

Discuss this question LIVE

11 mins ago

Text solutionVerified

There are 11 words

2 M^{s}, 2 A^{s}, 2 T^{s}, H, E, I, C, S

.
(i) Hence the number of words by taking all at a time.

= \frac{11 !}{2 ! 2 ! 2 !} = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 !}{2 \times 2 \times 2}

= 990 \times 7 \times 720 = 990 \times 5040 = 4989600

.

(ii) To begin with

C

.
Having fixed

C

at first place we have 100 letters in which 2 are

M^{s}, 2

are

A^{s}

and

2

are

T^{s}

and rest 4 different.
Hence the number of words will be

\frac{10 !}{2 ! 2 ! 2 !} = \frac{10 \times 9 \times 8 \times 7 \times 6 !}{2 \times 2 \times 2}

= 90 \times 7 \times 720 = 630 \times 720 = 453600

.

(iii) To begin with

T

.
having fixed

T

in the first place we will have only 10 letters our out of which 2 are

M^{s}

and

2

are

A^{s}

and rest six are

H, E, I, C, S

and

T

.
hence the number of words is

\frac{10 !}{2 ! 2 !} = 907200

(i.e. double part of (ii))

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Question Text	How many different arrangement can be made by using all the letters in the MATHEMATICS? How many of them begin with $C$ ? How many of them begin with $T$ ?
Answer Type	Text solution:1
Upvotes	150

How many different arrangement can be made by using all the letters in the MATHEMATICS?How many of them begin with C? How many of them begin with T?

Text solutionVerified

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How many different arrangement can be made by using all the letters in the MATHEMATICS?
How many of them begin with $C$ ? How many of them begin with $T$ ?