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Question

How many way can you arrange the letters of the word TREES if

(1) It must start with a consonant and end with a vowel?

(2) R must be in the middle?

(3) It begins with an E?

(4) It begin with exactly one E?

(5) consonants and vowels alternate?

(1) It must start with a consonant and end with a vowel?

(2) R must be in the middle?

(3) It begins with an E?

(4) It begin with exactly one E?

(5) consonants and vowels alternate?

2017-01-08 02:25:28

eve
1 Answer

For questions $1$ and $2$, use Permutations Calculator and generate questions based on the word 'TREES'. You will get detailed answers.

(3) Fix one 'E' at the first position. Remaining $4$ letters can be arranged in $4!=24$ ways.

Required number of ways $=24$

(4) As obtained above, $24$ arrangements possible with first letter as 'E'.

Now fix one 'E' at the fist position and other 'E' at the second position. Remaining $3$ letters can be arranged in $3!=6$ ways.

Hence, required number of ways

$=24-6=18$

(5) Arrange the $3$ consonants in $3!=6$ ways. These three consonants can be separated by placing two 'E's between adjacent consonants.

Required number of ways $=6$

(3) Fix one 'E' at the first position. Remaining $4$ letters can be arranged in $4!=24$ ways.

Required number of ways $=24$

(4) As obtained above, $24$ arrangements possible with first letter as 'E'.

Now fix one 'E' at the fist position and other 'E' at the second position. Remaining $3$ letters can be arranged in $3!=6$ ways.

Hence, required number of ways

$=24-6=18$

(5) Arrange the $3$ consonants in $3!=6$ ways. These three consonants can be separated by placing two 'E's between adjacent consonants.

Required number of ways $=6$