find the number of 4 letter word that can be made using the letters of RADIANCE.?

The word "RADIANCE" contains 2 "A" and other 5 different letters.

For 4-letter words with 2 "A", choose 2 letters out of the other 5 letters (C(5,2)), and arrange them into a letter (P(4,4)/2!).
Number of 4-letter words with 2 "A" = C(5,2) × P(4,4)/2! = 5!/2!3! × 4!/2 = 10 × 12 = 120

For 4-letter words with 1 "A", choose 3 letters out of the other 5 letters (C(5,3)) and arrange them into a letter (P(4,4)).
Number of 4-letter words with 1 "A" = C(5,3) × P(4,4) = 5!/3!2! × 4! = 10 × 24 = 240

For 4-letter words without "A", arrange 4 letters out of the other 5 letters into a word.
Number of 4-letter words without "A" = P(5,4) = 5!/1! = 120

Total number of 4-letter words = 120 + 240 + 120 = 480

RADIANCE has 8 letters. There are 2 A's
Assuming the letters don't repeat:
number of 4-letter words = 8P4 / 2! = 8!/((8-4)!2!) = 840

If the letters can repeat:
Number of 4 -letter words:
7x7x7x7
= 7^4
=2401

Break it up into two parts.

One part has no repeated letters. You have 7 choices for the first, 6 for the second, ....

The second part has two A's. There are 6C2 ways to pick the other two letters (since there are 7 different letters, and 8 in all: 2 A's and 6 single letters).

For each of those there are 4!/2! ways to rearrange the letters.

Can you work out the numbers from there?