From a standard 52-card deck, how many 5-card hands consist entirely of red cards?

In mathematics, permutation is known as the process of arranging a set in which all the members of a set are arranged into some series or order. The process of permuting is known as the rearranging of its components if the set is already arranged. Permutations take place, in more or less important ways, in almost every area of mathematics. They frequently appear when different commands on certain finite sets are considered.

What is a Combination?

A combination is an act of choosing items from a group, such that (not like permutation) the order of choice does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the union of n things taken k at a time without repetition  In combination you can select the items in any order. To those combinations in which re-occurrence is allowed, the terms k-selection or k-combination with replication are frequently used.

Permutation Formula

In permutation r things are selected from a set of n things without any replacement. In this order of selection matter.

ⁿP_r = (n!) / (n-r)!

Here,

n = set size, the total number of items in the set

r = subset size , the number of items to be selected from the set

Combination Formula

In combination r things are selected from a set of n things and where the order of selection does not matter.

ⁿC_r = n!/(n−r)!r!

Here, 

n = Number of items in set

r = Number of items selected from the set

From a standard 52-card deck, how many 5-card hands consist entirely of red cards?

Solution:

There are total 26 red card i.e., 13 hearts and 13 diamonds.

From 26 red cards, choose 5.

The answer is the binomial coefficient

(²⁶C₅) and you can read this as 26 choose 5.

So there are

(²⁶C₅) = 26! ⁄ 5!(26−5)!  

= 26! ⁄ 5!21!  

= 26×25×24×23×22×21! ⁄ 5×4×3×2×1×21!  

= 26×25×24×23×22 ⁄ 5×4×3×2  

= 26⁄2×25⁄5×24⁄12×23×22  

= 13×5×2×23×22 

= 13×10×23×22  

= 130×506 = 65,780

Possible 5-card hands consisting of only red cards.

Similar Questions

Question 1: From a standard 52 card deck, how many 6 card hands consist entirely of black cards?

Solution:

There are total 26 black cards i.e., 13 clubs and 13 spades.

From 26 black cards, choose 6.

The answer is the binomial coefficient

(²⁶C₆) and you can read this as 26 choose 6.

So there are

(²⁶C₆) = 26! ⁄ 6!(26−6)!  

= 26! ⁄ 6!20!  

= 26×25×24×23×22×21×20! ⁄ 6×5×4×3×2×1×20!  

= 26×25×24×23×22×21 ⁄ 6×5×4×3×2  

= 26⁄2×25⁄5×24⁄24×23×22×21⁄3

= 13×5×23×22×7=13×115×154  

= 1495×154 = 230,230  

Possible 6-card hands consisting of only black cards.

Question 2: From a standard​ 52-card deck, how many 2-card hands consist entirely of black ​cards?

Solution:

There are total 26 black cards i.e., 13 clubs and 13 spades.

From 26 black cards, choose 2.

The answer is the binomial coefficient

²⁶C₂ and you can read this as 26 choose 2.

So there are

²⁶C₂ = 26! ⁄ 2!(26−2)!  

= 26! ⁄ 2!24!  

= 26×25×24! ⁄ 2×1×24!  

= 26×25 ⁄ 2  

= 26⁄2×25  

= 13×25=13×25  

= 325

Possible 2-card hands consisting of only black cards.