A deck of cards contains 6 cards numbered from 1, 2, 3, 4, 5, and to 6

Alternative approach :

Let the three cards be a , b and c

So the question becomes
is a<b<c ?

Now, a can have any three of the operations < = > with b and b can have any two operations with c
So total options = 3*2
Favourable: a<b<c ... Only one outcome.
Probablity = 1/6
D

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Total number of possible arrangements of 3 cards = 6C3 *3!

Total number of ways in which 3 card are in the increasing order = 6C3 * 1

Probability = 1/3! = 1/6

The number of ways to select 3 cards form 6 where order matters is 6P3 = 6 x 5 x 4 = 120. Let’s use a table to list the number of ways in which 3 numbers are in increasing order if the first 2 numbers (in increasing order) are drawn:

First two numbers Number of ways
1, 2 4
1, 3 3
1, 4 2
1, 5 1
2, 3 3
2, 4 2
2, 5 1
3, 4 2
3, 5 1
4, 5 1

We see that there are 10 + 6 + 3 + 1 = 20 ways in which 3 numbers can be drawn without replacement such that they are in increasing order (notice that each addend is the number of ways where the smallest of the 3 numbers is 1, 2, 3 and 4, respectively). Therefore, the probability is 20/120 = 1/6.

Answer: D

[quote="nick1816"]Total number of possible arrangements of 3 cards = 6C3 *3!

Total number of ways in which 3 card are in the increasing order = 6C3 * 1

Probability = 1/3! = 1/6


How did u find the number of ways of drawing cards in increasing order as 6C3*1?

Hi riagarg07,

Let me try and explain.

From 6 cards if we were to select 3 cards we can do that in 6C3 or 20 ways. NOTE this will give us the total number of 3 card sets .e.g. 645, 321 are just two of such sets among the 20 sets.

Now in each of these sets only ONE ARRANGEMENT will be in ascending order.
e.g. 645 can be arranged in total 3! or 6 ways i.e.

456
654
645
546
465
565

Out of these only one set (456) is in ascending order.

Hence EVERY 3 card set will have only one way in which that set can be arranged in ascending order.
Now we have a total of 6C3 or 20 sets and in each set there is only one way in which that set can be in ascending order.

Hence total number of ways in which 3 cards can be chosen from 6 cards such that the cards are in ascending order is
6C3 *1

hope it's clear.

You can choose 3 cards from 6 cards and arrange them in 6P3 ways; i.e. 120 ways. (Permutations)

You can choose 3 cards from 6 cards and arrange them in one way, i.e. 6C3; i.e. 20 ways; that is any unique combination of three numbers will effectively be in increasing order.

Ans is 20/120 = 1/6 (Option D)

Hi Nick1816,
Can you please explain how have you counted the number in the increasing order as 6C3. Though I got the same result by counting the possible arrangements but I want to understand the thought process. Please explain.

Vinay

Hey
Suppose you choose 1, 2 and 3.

Total possible arrangements of 1, 2 and 3 are 3!= 6. [(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2) and (3,2,1)]

But there is only 1 possible arrangement out of those 6 arrangements in which you can put these numbers in increasing order, i.e (1,2,3), and this is true for any combination of 3 distinct numbers.


Hence, Total number of ways in which 3 card are in the increasing order = [number of ways to choose 3 numbers out of 6] * [possible arrangements of those 3 numbers]= 6C3 * 1

If you still have any doubt, you can ask.

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