Animated Solution for Mathematics - Permutations and Combinations: Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated are

Mastering Combinatorics

Imagine you are standing in front of a digital lock, tasked with forming a five-letter word from ten distinct letters. The problem seems simple enough, but then you encounter the phrase that strikes fear into the hearts of many students: 'at least one letter repeated.'

If you try to solve this by counting every possible scenario—one letter repeated, two letters repeated, three letters repeated—you will quickly find yourself drowning in a sea of cases. This is where the true beauty of combinatorics reveals itself: the Complementary Principle.

In JEE Advanced, whenever you see the phrase 'at least one', your first instinct should be to pause. Direct counting is often a trap designed to waste your time.

Instead, think of the 'Universal Set' of all possible words. If we want to find the number of words with at least one repetition, we can simply take the total number of possible words and subtract the words that have no repetitions. It is the ultimate shortcut, turning a complex, multi-case problem into a simple subtraction.

The Total Universe

Let's define our Universal Set. We have five empty slots to fill, and for each slot, we have ten choices. Because the problem allows for repetition in the total count, each slot is independent.

The first slot has $10$ choices, the second has $10$, and so on, for all five slots. Mathematically, this is expressed as:

This gives us $100,000$ total possible words. This is our 'Total Universe'—the set of all possible combinations, regardless of whether letters repeat or not.

The Forbidden Set

Now, we need to identify the 'Forbidden Set'—the words that have absolutely no repetitions. If we want no letters to repeat, we are essentially performing a permutation.

For the first slot, we have $10$ choices. Once we pick a letter, we cannot use it again, so the second slot has $9$ choices, the third has $8$, the fourth has $7$, and the fifth has $6$. This is the classic permutation formula, denoted as $P(10,5)$.

Calculating this, we get:

These are the words where every single letter is distinct.

The Final Victory

Now, we apply the Complementary Principle. We have our Total Universe ($100,000$) and our Forbidden Set ($30,240$).

By subtracting the Forbidden Set from the Total Universe, we are left with exactly the words that contain at least one repetition. The calculation is straightforward:

By using the Complementary Principle, we bypassed a massive, error-prone calculation and arrived at the answer with elegance and precision. This is the mindset of a JEE topper: not just working hard, but working smart.

Remember, in the world of combinatorics, sometimes the best way to find what you want is to calculate everything you don't want and subtract it from the whole. The final answer is 69,760. Keep practicing, stay curious, and never let the 'at least one' trap catch you off guard!