Question

There are 720 permutations of the digits 1, 2 ,3 , 4, 5, 6 suppose these permutations are arranged from smallest to largest numerical values beginning from 123456 and and ending with 654321.Whihc of the following is correct?

• A. Number on the 124^th position is 213564
• B. Number on the 267^th position is 321546
• C. Number on the 124^th position is 213546
• D. Number on the 267^th position is 321564

Answer 1

The Correct Option is A, B

To solve this problem, we need to find the permutation of the set of digits \( \{1, 2, 3, 4, 5, 6\} \) at specific positions when all permutations are arranged in ascending order. There are \( 6! = 720 \) such permutations.

Let's break it down using the concept of lexicographic order:

1. Each set of permutations where a particular starting digit is fixed contains \( 5! = 120 \) permutations (since the remaining five digits can be arranged in \( 5! \) ways).

2. We need to find out which digits start the permutations for the 124^th and 267^th positions.

3. Finding the 124^th permutation:

   • Permutations beginning with '1' occupy positions 1 to 120.
   • Thus, position 124 belongs to permutations starting with '2'.
   • Among permutations that start with '2', the first digit position (following '2') occupies positions 121 to 240.
   • Since we want the 124^th permutation, it is the 4^th in the '2xxxxx' series. Thus, consider '21xxxx'.
   • Each set in '21xxxx' occupies 24 permutations (as \( 4! = 24 \)).
   • It finally means having a different second digit leads us to choose '213xxx'.
   • The specific permutation for position 124 is '213564'.

4. Finding the 267^th permutation:

   • Continuing with '2xxxxx' permutations from above, those with '21xxxx' are at positions 121 to 144 and '23xxxx' are at positions 241 to 300.
   • Thus, position 267 falls under '321xxx'. (The second block is '321xxx', positions 241-300)
   • Position 267 is 27^th in the '32xxxx' series, making it '321xxx'.
   • Permutations directly follow a subset order until '321546'.

In conclusion, the results match the listed options and we have:

• The number at the 124^th position is 213564.
• The number at the 267^th position is 321546.