#### Question

A five-digit number is written down at random. The probability that the number is divisible by 5, and no two consecutive digits are identical, is

##### Options

\[\frac{1}{5}\]

\[\frac{1}{5} \left( \frac{9}{10} \right)^3\]

\[\left( \frac{3}{5} \right)^4\]

None of these

#### Solution

Let number be *abcde*

Case 1 : *e* = 0*a*, *b*, *c* can be filled in 9 × 9 × 9 ways*c* = 0 ⇒ 9 × 8 × 1 ways and *d* has 9 choices*c* ≠ 0 ⇒ (9 × 9 × 9 – 9 × 8 × 1) = 657

in the case *d* has 8 choices ⇒ 657 × 8

Total case = 9 × 8 × 1 × 9 + 657 × 8 ⇒ 5904

Case 2 : *e* = 5

If *c* = 5,

if *a* ≠ 5 then *a*, *b*, *c* can be filled in 8 × 8 × 1 = 64 ways

if *a* = 5 then *a*, *b*, *c* can be filled in 1 × 9 × 1 = 9 ways

if *c* ≠ 5, then first 3 digits can be filled in 729 – 64 – 9 = 656 ways

here *d* has 8 choices

No. of member ending in 5 and no two consecutive digits being identical ⇒ (64 + 9) × 9 + 656 × 8

Hence, None of these