Factorial and gamma functions in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of factorial for a positive integer

What is the output of this code snippet using scipy's factorial function?

SciPy

from scipy.special import factorial
result = factorial(5, exact=True)
print(result)

A 120

B 24

C 720

D Error

💡 Hint

Recall that factorial of 5 is 5*4*3*2*1.

The factorial of 5 is 120 because 5*4*3*2*1 equals 120. Using exact=True returns an integer.

❓ data_output

intermediate

Gamma function values for half-integers

What is the output array from this code using scipy's gamma function for half-integers?

SciPy

import numpy as np
from scipy.special import gamma
x = np.array([0.5, 1.5, 2.5])
result = gamma(x)
print(result)

A [0.88622693 1.32934039 3.32335097]

B TypeError

C [1.0 1.0 1.0]

D [1.77245385 0.88622693 1.32934039]

💡 Hint

Gamma(0.5) equals sqrt(pi).

Gamma(0.5) is sqrt(pi) ≈ 1.772, Gamma(1.5) = 0.5*sqrt(pi) ≈ 0.886, Gamma(2.5) = 1.5*0.5*sqrt(pi) ≈ 1.329.

❓ visualization

advanced

Plotting factorial vs gamma function

Which option produces the correct plot comparing factorial and gamma functions for integers 1 to 6?

SciPy

import matplotlib.pyplot as plt
from scipy.special import factorial, gamma
import numpy as np
x = np.arange(1,7)
plt.plot(x, factorial(x, exact=True), 'o-', label='factorial')
plt.plot(x, gamma(x+1), 'x--', label='gamma(x+1)')
plt.legend()
plt.show()

A Plot shows two overlapping lines with factorial and gamma(x+1) values identical for integers 1 to 6.

B Plot shows factorial values increasing but gamma values constant at 1.

C Plot shows factorial values decreasing and gamma values increasing.

D Plot raises a TypeError due to incompatible types.

💡 Hint

Recall gamma(n+1) = n! for positive integers n.

Gamma function shifted by 1 matches factorial exactly for positive integers, so plots overlap.

🧠 Conceptual

advanced

Understanding gamma function domain

Which statement about the gamma function domain is correct?

A Gamma function is defined for all real numbers including zero.

B Gamma function is defined for all real numbers except non-positive integers.

C Gamma function is only defined for positive integers.

D Gamma function is only defined for complex numbers with positive real part.

💡 Hint

Think about poles of the gamma function.

Gamma function has poles at zero and negative integers, so it is undefined there but defined elsewhere.

🔧 Debug

expert

Identify error in gamma function usage

What error does this code raise?

SciPy

from scipy.special import gamma
result = gamma(-2)
print(result)

A ValueError: math domain error

B ZeroDivisionError

C No error, prints inf

D OverflowError

💡 Hint

Gamma has poles at negative integers, check what scipy returns.

Gamma at negative integers returns infinity (inf) in scipy, no exception is raised.

Practice

1. What does the factorial function calculate for a positive integer n?

easy

A. The number of ways to arrange n items in pairs

B. The sum of all positive integers from 1 to n

C. The product of all positive integers from 1 to n

D. The square of n

Solution

1. Step 1: Understand factorial definition

   Factorial of n is the product of all positive integers from 1 up to n.

2. Step 2: Compare options with definition

   Only The product of all positive integers from 1 to n matches this definition exactly.

3. Final Answer:

   The product of all positive integers from 1 to n -> Option C

4. Quick Check:

   Factorial(n) = 1*2*...*n [OK]

Hint: Factorial multiplies numbers up to n, not adds [OK]

Common Mistakes:

• Confusing factorial with sum
• Thinking factorial is n squared
• Mixing factorial with combinations

2. Which of the following is the correct way to import the factorial function from scipy.special in Python?

easy

A. from scipy.special import factorial

B. import scipy.factorial

C. from scipy import factorial

D. import factorial from scipy.special

Solution

1. Step 1: Recall correct import syntax

   In Python, to import a specific function from a module, use from module import function.

2. Step 2: Match with scipy.special factorial

   The factorial function is inside scipy.special, so correct import is from scipy.special import factorial.

3. Final Answer:

   from scipy.special import factorial -> Option A

4. Quick Check:

   Correct import syntax = from scipy.special import factorial [OK]

Hint: Use 'from module import function' for specific imports [OK]

Common Mistakes:

• Using 'import scipy.factorial' which is invalid
• Trying 'from scipy import factorial' missing 'special'
• Incorrect order like 'import factorial from ...'

3. What is the output of the following code?

from scipy.special import factorial
result = factorial(5, exact=True)
print(result)

medium

A. 24

B. 5

C. 720

D. 120

Solution

1. Step 1: Understand factorial(5) calculation

   Factorial of 5 is 1*2*3*4*5 = 120.

2. Step 2: exact=True returns integer factorial

   Using exact=True returns the exact integer value, so result is 120.

3. Final Answer:

   120 -> Option D

4. Quick Check:

   factorial(5) = 120 [OK]

Hint: Factorial 5 is 120, exact=True returns integer [OK]

Common Mistakes:

• Confusing factorial(5) with factorial(4)
• Ignoring exact=True and expecting float
• Mistaking factorial for sum

4. Identify the error in this code snippet:

from scipy.special import gamma
print(gamma(0))

medium

A. gamma(0) causes a division by zero error

B. gamma function requires integer input only

C. Missing import for factorial function

D. No error, output is 1

Solution

1. Step 1: Understand gamma function behavior at zero

   The gamma function is undefined at zero and negative integers, causing a division by zero error.

2. Step 2: Check code output

   Calling gamma(0) raises a runtime warning or error due to singularity.

3. Final Answer:

   gamma(0) causes a division by zero error -> Option A

4. Quick Check:

   gamma(0) undefined = error [OK]

Hint: Gamma undefined at zero and negatives [OK]

Common Mistakes:

• Assuming gamma accepts only integers
• Expecting gamma(0) to be 1
• Ignoring runtime warnings

5. You want to calculate the factorial of 4.5 using scipy. Which function and approach should you use to get a valid result?

hard

A. Use factorial(4.5, exact=True) directly

B. Use gamma(5.5) because gamma(n) = (n-1)! for real numbers

C. Use factorial(4) and ignore decimals

D. Use gamma(4.5) directly

Solution

1. Step 1: Understand factorial vs gamma for non-integers

   scipy.special.factorial(exact=True) is defined for non-negative integers only, but gamma extends factorial to real numbers with gamma(n) = (n-1)!.

2. Step 2: Calculate factorial of 4.5 using gamma

   To get factorial of 4.5, use gamma(4.5 + 1) = gamma(5.5).

3. Final Answer:

   Use gamma(5.5) because gamma(n) = (n-1)! for real numbers -> Option B

4. Quick Check:

   factorial(4.5) = gamma(5.5) [OK]

Hint: Use gamma(n+1) for factorial of non-integers [OK]

Common Mistakes:

• Trying factorial(4.5, exact=True) on non-integers
• Using gamma(n) instead of gamma(n+1)
• Ignoring decimal part and truncating