Answer :

**Given****:** the word is ‘SUNDAY.’

**To find:** number of words that can be formed with the letters of the given word, that can begin with N, and that can begin with N and end in Y

Total number of letters = 6

**Formula used:**

Number of arrangements of n things taken all at a time = P(n, n)

∴ Total number of arrangements

= the number of arrangements of 6 things taken all at a time

= P(6, 6)

{∵ 0! = 1}

= 6!

= 6 × 5 × 4 × 3 × 2 × 1

= 720

**Hence, the total number of words can be made by letters of the word ‘SUNDAY’ = 720**

Now, we need to find out of a number of words starting with N

So, fix the position of the first letter as N:

Remaining number of letters in the word ‘SUNDAY’ = 5

Now, we need to arrange these 5 letters at 5 places.

**Formula used:**

Number of arrangements of n things taken all at a time = P(n, n)

∴ The total number of ways

= the number of arrangements of 5 things taken all at a time

= P(5, 5)

{∵ 0! = 1}

= 5!

= 5 × 4 × 3 × 2 × 1

= 120

**Hence, the possible number of words using letters of ‘SUNDAY’ starting with ‘N’ is 120**

Now, we need to find out a number of words starting with N and ending with Y

So, fix the position of first and last letter as N and Y:

Remaining number of letters = 4

Now, we need to arrange these 4 letters at 4 places.

**Formula used:**

Number of arrangements of n things taken all at a time = P(n, n)

∴ The total number of ways

= the number of arrangements of 4 things taken all at a time

= P(4, 4)

{∵ 0! = 1}

= 4!

= 4 × 3 × 2 × 1

= 24

**Hence, the possible number of words using letters of ‘SUNDAY’ starting with ‘N’ and ending with ‘Y’ are 24**

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