In How Many Different Ways Can The Letters Of The Word 'mathematics'

1.	In How Many Different Ways Can The Letters Of The Word 'mathematics' Be Arranged So That The Vowels Always Come Together?
Answer» In the word 'MATHEMATICS', we treat the vowels AEAI as one letter. THUS, we have MTHMTCS (AEAI). Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different. So,Number of ways of arranging these letters = (8!/2!2!)=10080. Now, AEAI has 4 letters in which A occurs 2 TIMES and the rest are different. Number of ways of arranging these letters = (4!/2)=12. => REQUIRED number of words = (10080 12) = 120960. In the word 'MATHEMATICS', we treat the vowels AEAI as one letter. Thus, we have MTHMTCS (AEAI). Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different. So,Number of ways of arranging these letters = (8!/2!2!)=10080. Now, AEAI has 4 letters in which A occurs 2 times and the rest are different. Number of ways of arranging these letters = (4!/2)=12. => Required number of words = (10080 12) = 120960.

In How Many Different Ways Can The Letters Of The Word 'mathematics' Be Arranged So That The Vowels Always Come Together?

Answer»

In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.

THUS, we have MTHMTCS (AEAI).

Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.

So,Number of ways of arranging these letters = (8!/2!*2!)=10080.

Now, AEAI has 4 letters in which A occurs 2 TIMES and the rest are different.

Number of ways of arranging these letters = (4!/2)=12.

=> REQUIRED number of words = (10080 * 12) = 120960.

In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.

Thus, we have MTHMTCS (AEAI).

Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.

So,Number of ways of arranging these letters = (8!/2!*2!)=10080.

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.

Number of ways of arranging these letters = (4!/2)=12.

=> Required number of words = (10080 * 12) = 120960.