Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from $\{\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\}$ 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 $\alpha \times 5^{6}$, then $\alpha$ is equal to ____.

JEE Main · 2022 · 28 Jul 2022 · Shift 1 · Permutations and Combinations

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}

α \times 5^{6}

, then

α

is equal to ____.

Answer: 7073

Total characters available: 10 (5 alphabets + 5 numbers).

Total passwords =

1 0^{6} + 1 0^{7} + 1 0^{8}

(for lengths 6, 7, 8).

Passwords with only alphabets =

5^{6} + 5^{7} + 5^{8}

.

Passwords with at least one number =

(1 0^{6} - 5^{6}) + (1 0^{7} - 5^{7}) + (1 0^{8} - 5^{8})

= 1 0^{6} (1 + 10 + 100) - 5^{6} (1 + 5 + 25)

= 1 0^{6} \cdot 111 - 5^{6} \cdot 31

= 2^{6} \cdot 5^{6} \cdot 111 - 5^{6} \cdot 31

= 5^{6} (64 \cdot 111 - 31)

= 5^{6} (7104 - 31)

= 5^{6} \cdot 7073

Therefore

α = 7073

JEE Main · 2022 · 28 Jul 2022 · Shift 1 · Integer answer