“Choose” means “select”. “n choose k formula” is used to find the number of ways selecting k things out of n things. Sometimes we prefer selecting but we do not care about ordering them. For example, you are packing luggage for a trip, and you want to carry 5 dresses out of 10 new dresses you have. This can be done in various ways, it doesn’t matter in which order you have chosen them but it all matters what dresses you have chosen. Let us learn the n choose k formula along with a few solved examples.
What is n Choose k Formula?
It is also known as a binomial coefficient. It is used to find the number of ways of selecting k different things from n different things. The n choose k formula is also known as combinations formula (as we call a way of choosing things to be a combination). This formula involves factorials.
The n Choose k Formula is:
C (n , k) = n! / [ (n-k)! k! ]
Solved Examples Using n Choose k Formula
Example 1
Find the number of ways of forming a team of 5 members out of 10 members.
Solution
To find: The number of ways of forming a team of 5 members out of 10 members.
Here, the total number of members is n = 10.
The number of members to be selected to form a team is k = 5.
This can be done in C(n , k) ways and we will find this using the n choose k formula.
C (n , k) = n! / [ (n-k)! k! ]
C (10, 5) = 10! / [ (10 – 5)! 5! ]
C(10, 5) = (10 × 9 × 8 × 7 × 6 × 5!) / (5! 5!)
C(10, 5) = (10 × 9 × 8 × 7 × 6) / (5 × 4 × 3 × 2 × 1)
C(10, 5) = 252.
Answer: The number of ways of forming a team of 5 members = 252.
Example 2
There are 50 people at a party. Among them, each person shakes hand with each of the others exactly once. Then how many handshakes are supposed to happen there?
Solution
To find: The total number of shake hands at the party.
The total number of people at the party is n = 50.
For a shaking hand to happen, there must be two people.
So the number of shake hands is equal to the number of ways of selecting 2 people out of 50 people. So k = 2.
Using the n choose k formula,
C (n , k) = n! / [ (n-k)! k! ]
C (50, 2) = 50! / [ (50 – 2)! 2! ]
C (50, 2) = (50 × 49 × 48!) / (48! 2!)
C (50, 2) = (50 × 49) / (2 × 1)
C (50, 2) = 1225.
Answer: The number of shake hands = 1,225.
N Choose K Formula
N choose K is called so because there is (n/k) number of ways to choose k elements, irrespective of their order from a set of n elements.
To calculate the number of happenings of an event, N chooses K tool is used. This is also called the binomial coefficient.
The formula for N choose K is given as:
C(n, k)= n!/[k!(n-k)!]
Where,
n is the total numbers
k is the number of the selected item
Solved Example
Question: In how many ways, it is possible to draw exactly 6 cards from a pack of 10 cards?
Solution:
From the question, it is clear that,
n = 10
k = 6
So the formula for n choose k is,
C(n, k)= n!/[k!(n-k)!]
Now,
A bit of theory – foundation of combinatorics
Variations
A variation of the k-th class of n elements is an ordered k-element group formed from a set of n elements. The elements are not repeated and depend on the order of the group’s elements (therefore arranged).
The number of variations can be easily calculated using the combinatorial rule of product. For example, if we have the set n = 5 numbers 1,2,3,4,5 and we have to make third-class variations, their V3 (5) = 5 * 4 * 3 = 60.
n! we call the factorial of the number n, which is the product of the first n natural numbers. The notation with the factorial is only clearer, equivalent. For calculations, it is fully sufficient to use the procedure resulting from the combinatorial rule of product.
Permutations
The permutation is a synonymous name for a variation of the nth class of n-elements. It is thus any n-element ordered group formed of n-elements. The elements are not repeated and depend on the order of the elements in the group. P(n)=n(n−1)(n−2)…1=n! A typical example is: We have 4 books, and in how many ways can we arrange them side by side on a shelf?
Variations with repetition
A variation of the k-th class of n elements is an ordered k-element group formed of a set of n elements, wherein the elements can be repeated and depends on their order. A typical example is the formation of numbers from the numbers 2,3,4,5, and finding their number. We calculate their number according to the combinatorial rule of the product: Vk′(n)=n⋅n⋅n⋅n…n=nk
Permutations with repeat
A repeating permutation is an arranged k-element group of n-elements, with some elements repeating in a group. Repeating some (or all in a group) reduces the number of such repeating permutations.
A typical example is to find out how many seven-digit numbers formed from the numbers 2,2,2, 6,6,6,6.
Combinations
A combination of a k-th class of n elements is an unordered k-element group formed from a set of n elements. The elements are not repeated, and it does not matter the order of the group’s elements. In mathematics, disordered groups are called sets and subsets. Their number is a combination number and is calculated as follows:
A typical example of combinations is that we have 15 students and we have to choose three. How many will there be?
Combinations with repeat
Here we select k element groups from n elements, regardless of the order, and the elements can be repeated. k is logically greater than n (otherwise, we would get ordinary combinations). Their count is:
Explanation of the formula – the number of combinations with repetition is equal to the number of locations of n − 1 separators on n-1 + k places. A typical example is: we go to the store to buy 6 chocolates. They offer only 3 species. How many options do we have? k = 6, n = 3.
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