Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A, B, C, D, E} or a number from {1,2,3,4,5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1,2,3,4,5} is α × 5⁶, then α is equal to ___________.

Solution

Maths sir · 4 steps

"At least one number" means count all passwords, then remove the ones made only of letters.

1. Use the complement. Each position has 10 choices (5 letters and 5 numbers), so there are 10ⁿ passwords of length n. The ones with no number use only the 5 letters, and there are 5ⁿ of those. So 10ⁿ - 5ⁿ passwords of length n contain at least one number.
2. Pull out the power. Since 10ⁿ = 5ⁿ × 2ⁿ, this count is 5ⁿ(2ⁿ - 1). Add the lengths 6, 7 and 8, writing 5⁷ = 5 × 5⁶ and 5⁸ = 25 × 5⁶.

   Σₙ₌₆⁸ 5ⁿ(2ⁿ-1) = 5⁶(63 + 5 × 127 + 25 × 255)

3. Add up. Inside the bracket: 63 + 635 + 6375 = 7073.
4. So. The number of such passwords is 7073 × 5⁶, so α = 7073.

The idea to keep. Whenever a question says "at least one", count everything and subtract the cases with none. Then factor out the common power so the answer matches the asked form.

Answer from the NTA final answer key. Solution by PrepWiser.