Question

The number of ways in which four letters can be selected from the word 'DEGREE', is

• A. 7
• B. 6
• C. \(\frac{6!}{3!}\)
• D. None of these

Hint:

Handle selections with identical objects by considering cases based on the number of identical letters chosen.

Answer 1

The Correct Option is A

To determine the number of ways to select four letters from the word "DEGREE", we need to account for the repetition of letters in the word. The word consists of the letters: D, E, G, R, E, E.

First, let's identify the unique letters and their frequencies:

• D: 1
• E: 3
• G: 1
• R: 1

Since the letter 'E' repeats three times, and we have a total of 6 letters, we can approach the selection in cases based on the number of 'E's selected:

1. Selecting four letters with fewer than three 'E's:
   • The possible scenarios could be: 0 'E', 1 'E' or 2 'E's.
2. Case 1: Selecting 0 'E's:
   • Combination of D, G, and R (there are only three letters to choose from), so it's not possible to select 4 letters. Therefore, this case does not contribute to the total.
3. Case 2: Selecting 1 'E':
   • We choose 3 other letters from D, G, R and two other 'E's. Therefore, selecting 3 more letters from D, G, R (3C3 = 1).
4. Case 3: Selecting 2 'E's:
   • We choose 2 letters from D, G, R. Combination (3C2 = 3).
5. Selecting three or all 'E's:
   • Case 4: Selecting 3 'E's: We need one more letter from D, G, R, so it's just 3 combinations.

Adding all possible selections:

1 (3 non-E selections) + 3 (two E selections) + 3 (one E selection) = 7 total ways to select four letters from "DEGREE".

Therefore, the correct answer is: 7.