AMC 10 Weekly Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 10 Medium

Sammy uses a permutation of the six digits 1, 4, 0, 3, 2, 2 as his six-digit bank card password. If there is exactly one digit between the two 2’s, and 1 and 4 are adjacent, then the number of possible passwords is （   ）．

  • A.

    48

  • B.

    32

  • C.

    24

  • D.

    16

  • E.

    8

Answer:C

The digits 1 and 4 must be adjacent, which gives _{2}P_{2} = 2 arrangements.

 

The two 2’s must have exactly one digit between them, which gives _{2}P_{1} = 2 arrangements.

 

After grouping these conditions together, the combined blocks can be permuted in _{3}P_{3} = 6 ways.

 

Therefore, the total number of possible passwords is 2 \times 2 \times 6 = 24.

 

Thus, the answer is \text{C}.

Related Problems
Problem 8 AMC 10 Weekly Practice Round 3
Problem 9 AMC 10 Weekly Practice Round 3
Problem 11 AMC 10 Weekly Practice Round 3
Problem 12 AMC 10 Weekly Practice Round 3
Think Academy Powered by ChatGPT