Question

There are 720 permutations of the digits 1,2,3,4,5 and 6. Suppose these permutations are arranged from smallest to largest numerical values, beginning from 123456 and ending with 654321. What is the position of the number 321546?
A. 165
B. 123
C. 267
D. 763

Answer 1

Hint: Here numbers are arranged from smallest to largest, so we will fix the smaller number at first position . After fixing 1 at first position, arrange the remaining number using permutations and combinations.

Complete step-by-step answer:
Given, there are 720 permutation of the digits 1,2,3,4,5 and 6
Also, if we suppose that they are arranged from smallest to largest numerical values, beginning from 123456 and ending with 654321
Then the smallest numerical values will have 1 fixed in the first place from left and therefore there are 5! =120 Possibilities to arrange the other five places.
So, it is decided that up to 120th place numbers beginning with the smallest number 1.
Now the 121th number will start from 2 hence from this place we will have 5! =120 numbers starting from 2.
Therefore, we have 120+120=240 numbers fixed from lowest to highest
Now from 241th position the numbers will start from 3 and that would be 312456
And if we fix 31 _ _ _ _ then we have 4! =24 Possibilities for other digits
Which implies we have fixed number up to 264th position
Now 265th position number will be 321456
266th position number will be 321465
And 267th position number would be 321546
Therefore, option C is correct.

Note: These types of questions can be easily solved using the basic concept of fixing a digit and how other digits can be arranged. Combination is a selection of items from a collection, such that the order of selection does not matter