NIMCET 2020 — Mathematics PYQ

Step 1: Analyze the given word.

The given word is DELHI.

• Total number of letters = $5$ (which are D, E, L, H, I).

• Notice that all $5$ letters are distinct (none of them are repeated).

Step 2: Apply the given condition.

The question specifies that the newly formed words must start with the letter D, and letters cannot be repeated.

Let's represent the 5-letter word positions as:

$$\text{D }$$

• The $1^{\text{st}}$ position is fixed with the letter D (only $1$ way).

• We are now left with $4$ remaining distinct letters (E, L, H, I) to fill the remaining $4$ positions.

Step 3: Calculate the permutations of the remaining letters.

The number of ways to arrange $4$ distinct objects into $4$ vacant spots is given by $4!$ (4 factorial):

$$\text{Total arrangements}=4!$$

$$4!=4\times 3\times 2\times 1=24$$

Thus, there are $24$ different words that can be formed starting with the letter D.

Correct Option: C (24)