Question

When she chooses a password, Sandy always uses exactly ten different characters: five letters (A, B, C, D, and E) and five numbers (2, 3, 4, 5, and 6). Additionally, she always ensures that no pair of letters is consecutive and that no pair of numbers is consecutive. The number of different passwords which can be formed with the above rules is:

• A. Fewer than 1,000
• B. Between 1,000 and 10,000
• C. Between 10,000 and 100,000
• D. Between 100,000 and 1,000,000
• E. More than 1,000,000

Answer 1

The correct option is C Between 10,000 and 100,000

(1) First arrange letters $A,B,C,D,E$ in 5 plaus in $5!$ ways.

Then arrange numbers in between

$\Rightarrow A\_B\_C\_D\_E\_$

$\Rightarrow 5!\times 5!=14400$

(2) Arrange numbers in $5!$ ways. then arrange letters in between them.

$1\_2\_3\_4\_5\_$

$\Rightarrow 5!\times 5!=14400$

So, in both the cases we get $14400$ no. of possible ways.

Thus, the total number of ways $=14400+14400=28800$

Hence, the number of different passwords formed lies between $10,000$ and $100,00$.