Combinations - Lab

Introduction

Now, let's dive into combinations. In the previous lab, you saw how the order was important when using permutations. Cracking a code is one example, but what if the order doesn't matter, for example, when an engaged couple wants to pick 3 wedding cakes from a list of 15? You'll need to use another technique here, and this is where combinations come in handy!

Objectives

You will be able to:

• Decide whether or not permutations and combinations are required for a given problem
• Use Python to calculate combinations and permutations

Let's get started

From the previous lab, you remember that we created a factorial function.

Now, let's use this factorial function to create a function combination as well as permutation, both holding 2 arguments n and k.

def factorial(n):
    prod = 1
    while n >= 1:
        prod = prod * n
        n = n - 1
    return prod

def permutation(n,k):
    None

def combination(n,k):
    None

Great! We can use these functions in the following exercises.

Permutations or Combinations?

Flatiron School is holding a mini mathematics contest and there are 9 people in the last round.

a. Imagine flatiron school is giving out bronze, silver, and gold medal respectively. How many possible ways are there to create this top three?

medal_top_3 = None
medal_top_3 # 504.0

b. Imagine Flatiron school granting the first three contestants a massive fruit basket. How many ways of selecting three people are there in this case?

scholarship_top_3 = None
scholarship_top_3 # 84.0

Some More Practice using Combinations

Imagine you have 6 different consonants and 4 different vowels written on pieces of paper in a bag. You'll draw 5 letters out of the bag.

a. What is the probability that you draw exactly 2 consonants and 3 vowels when drawing 5 letters?

Write the code for getting total number of ways of drawing 2 out of 6 and 3 out of 4 below

draw_cons = None
draw_vow = None

The total number of ways to draw 5 letters out of 10 letters.

sample = None

The probability of drawing 2 consonants and 3 vowels when drawing 5 letters:

None # 0.23809523809523808

b. Out of 6 consonants and 4 vowels, how many words with 2 consonants and 3 vowels can be formed?

You can reuse a part of the previous exercise. Which part? print the result below.

draw_cons = None
draw_vow = None

Now we need to take into account that order is important.

order_5_letters = None

The total number of words with 2 consonants and 3 vowels then equals:

total_words = None
print("In total,",  total_words, "words with 2 consonants and 3 vowels can be formed from our existing letter pool.")
# In total, 7200.0 words with 2 consonants and 3 vowels can be formed from our existing letter pool.

Combinations: Creating Soccer Teams

We're holding a mini soccer tournament and 16 people are participating. We'd like to form 4 teams of 4. How many ways are there to do this?

# your code here  # the answer is 63063000.0

Summary

In this lab, you got some practice with combinations, and deciding whether or not combinations and permutations are required for a problem. Congrats! Combinations and permutations are the cornerstones of combinatorics, and you now know how to use Python to compute them in various settings.