Question

Find the number of 4 letter words that can be formed using the letters of the word EQUATION.How many of them begin with the letter E? How many end with letter N?How many begin with E and end with N?

Answer 1

To Find :-

• Number of letters form by using any four alphabet of EQUATION word.

• Number of letter starting with E.

• Number of letter ending with N.

• Number of letter starting with E and ending with N .

Solution :-

1st case If repetition is allowed :-

We have total letters = 8

We have places = $\square \: \square \: \square \: \square$

$\square \: \: \: \: \: \square \: \: \: \square \: \: \: \: \square \\ 8 \times 8 \times 8 \times 8$

• $\sf{\implies 4096\ letters }$

If repetition is not allowed :-

$\square \: \: \: \: \: \square \: \: \: \square \: \: \: \: \square \\ 8 \times 7 \times 6 \times 5$

• $\sf{\implies 1680\ letters }$ 2nd case Letters beginning with E.

If repetition is allowed

$\square \: \: \: \: \: \square \: \: \: \square \: \: \: \: \square \\ 1(E) \times 8 \times 8 \times 8$

• $\sf{\implies 512\ letters }$

If repetition is not allowed

$\square \: \: \: \: \: \square \: \: \: \square \: \: \: \: \square \\ 1(E) \times 7 \times 6 \times 5$

• $\sf{\implies 210\ letters }$

Case 3rd Letter ending with N .

If repetition is not allowed :-

$\square \: \: \: \: \: \square \: \: \: \square \: \: \: \: \square \\ 5 \times 6 \times 7 \times 1(N)$

• $\sf{\implies 210\ letters }$

If repetition is allowed :-

$\square \: \: \: \: \: \square \: \: \: \square \: \: \: \: \square \\ 8 \times 8 \times 8 \times 1(N)$

• $\sf{\implies 512\ letters }$

4th case Start with E and with N

If repetition is allowed :-

$\square \: \: \: \: \: \square \: \: \: \square \: \: \: \: \square \\ 1(E) \times 8 \times 8 \times 1(N)$

• $\sf{\implies 64\ letters }$

If repetition is not allowed :-

$\square \: \: \: \: \: \square \: \: \: \square \: \: \: \: \square \\ 1(E) \times 6 \times 5 \times 1(N)$

• $\sf{\implies 30\ letters }$

Answer 2

To Find :-

Number of letters form by using any four alphabet of EQUATION word.

Number of letter starting with E.

Number of letter ending with N.

Number of letter starting with E and ending with N .

Solution :

CASE 1

In the word EQUATION we have letters = 8

We have places = 4

n = 8  , r = 4

Using formula nPr

8P4 = 1680

CASE 2

begin with letter E

We have places = 3

We have Letters = 7

7P3  =  210

CASE 3

begin with letter N

We have places = 3

We have Letters = 7

7P3  =  210

CASE 4

Begin with E and end with N

We have places = 2

We have letters  = 6

6P2 = 30