Question

The P₃⁴ (= 24) 3-permutations of the set {1, 2, 3, 4} can be arranged in the following way, called the lexicographic ordering: 123, 124, 132, 134, 142, 143, 213, 214, 231, 234, 241, 243, 312, · · · , 431, 432. Thus the 3-permutations “132” and “214” appear at the 3rd and 8th positions of the ordering respectively. Now, there are P₄⁹(= 3024) 4-permutations of the set {1, 2, · · · , 9}. What are the positions of the 4-permutations “4567” and “5182” in the corresponding lexicographic ordering of the 4-permutations of {1, 2, · · · , 9}.

Answer 1

We have total 3024 permutations.

By fixing the first number->Each number would have 8P3 permutations=336

By fixing first and second-> each would have 7P2 permutations=42

By fixing first, second and third-> each would have 6P1 permutations=6

For 4567->First all 1,2,3 should be over, Then comes 4.

For second digit->1,2,3 should be over. Then comes 5.

For third digit->1,2,3 should be over. Then comes 6

For fourth digit->1,2,3 should be over. Then comes 7

Rank=3*336+3*42+3*6+4=1156

For 5182-> First all 1,2,3,4 should be over. Then comes 5

Second digit->1

Third digit-> 2,3,4,6,7 should be over. THen comes 8

Fourth digit-> Direct 2

Rank->336*4+6*5+1=1375