We all have tried adding numbers from 1 with their consecutive numbers, like 1 + 2 + 3 + 4 + 5 which is 15. Some may have tried multiplication too. 1 × 2 × 3 × 4 × 5 = 120. It is easier to start from 5 and go backwards to 1 to do the multiplication. The multiplication of consecutive numbers is called the Factorials. Here 1 × 2 × 3 × 4 × 5 is represented as 5!. Factorials is a mathematical function that is used to represent, arrangement of things. Factorials are used in topics like Permutations and combinations, series, sequences and many more.
Factorial questions with solutions help the students to understand the topic clearly by the method of practice. By solving the varying difficulty levels in the questions, builds the confidence of the students and reduces the fear of maths problems. You may also refer to Factorial formula and examples.
Factorial Questions With Solutions
Here are a few Factorial questions with steps that lead to the answer.
Question 1:
What is the value of 7!
Solution:
We know that n! = 1 × 2 × 3 × …. (n-1)! × n!.
Here n= 7, Therefore n! = 7!
7! = 1 × 2 × 3 × 4 × 5 × 6 × 7
7! = 120 × 6 × 7
7! = 120 × 42
7! = 5040
Question 2:
If (n+1)! = 5n!, solve for n.
Solution:
Given (n+1)! = 5n!
We know that by the factorial formula, n! = n! × (n-1)!
Therefore (n+1)! = (n+1) × (n+1 -1)! (Because here n = n + 1)
But (n+1)! = 5n!
Substituting
5n! = (n+1) × (n+1 -1)!
5n! = (n+1) × (n)!
n! Gets cancelled at both LHS and RHS
5 = n+1
5-1 = n
n = 4.
So n = 4 in the problem (n+1)! = 5n!
Question 3:
Solve for (9! × 8!) /(7! × 6!)
Solution:
This can be solved by 2 methods, 1 by expanding every factorial by then multiplying and dividing. The 2nd method, is shown below
Given (9! × 8!) /(7! × 6!)
We can write
(9! × 8!) = (9 × 8 × 7!) × (8 × 7 × 6!)
Substituting
(9 × 8 × 7!) × (8 × 7 × 6!)
———————————–
7! × 6!
9 × 8 7! × 8 × 7 × 6!
———————————-
7! × 6!
Cancelling 7! And 6! From both numerator and denominator
9 × 8 × 8 × 7
= 72 × 56
= 4032
Therefore (9! × 8!) /(7! × 6!) = 4032
Question 4:
Prove that 0! is 1
Solution:
Let us write down the factorial formula
n! = n(n-1)!
Let us find the value when n = 1
1! = 1(1-1)!
1= 1.(0!)
1 = 0!
Or 0! = 1.
Proved
A more complex topic on factorials can be found at exponents of a prime factorial.
Question 5:
Simplify (6! × 3!)/(3!)
​​Solution:
Given (6! × 3!)!/3!!
3! Gets cancelled as both numerator and denominator have 3!
Therefore the answer to the given problem is 6! which is 6 × 5!
But 5! = 120
So 6! = 6 × 5! = 6 × 120 = 720.
Question 6:
Solve n!/r!(n-r)! When n = 9 and r = 8
Solution:
Given n = 9 and r = 8
n! = 9!
—- ———
r!(n-r)! 8! (9-8)!
= 9!
——-
8! × 1!
= 9 × 8!
————-
8!
= 9 . (8! Get cancelled as it is present at both numerator and denominator)
Therefore n!/r!(n-r)! = 9, when n = 9 and r = 8
Question 7:
How many prime divisors does 3! have?
Solution:
Given 3!
3! = 3 × 2 × 1 = 6
3! has 2 prime divisors, which are 3 and 2.
Question 8:
If n! = 24, find n
Solution:
We know that by the formula of factorial n! = 1 × 2 × 3 × ….(n-1) × n!
Given n! = 24
1 × 2 × 3 × 4… n! = 24
n = 4.
Question 9:
Solve for x2 -4x+4 = 0, and provide the answer in factorial.
Solution:
Given x2-4x+4 = 0
x2-2x-2x+4 = 0
x(x-2)-2(x-2) = 0
(x-2) (x-2) = 0
Therefore x = 2
Or x= 2!. Because 2! = 2 × 1 = 2
Therefore for the equation x2 -4x+4 = 0, the answer in terms of factorial is 2!
Question 10.
Can you prove that 10 ! = 8! +2!
Solution:
10! will be equal to 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3628800
8! will be 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40320
2! = 2 × 1 = 2
To prove that 10! = 8! + 2!
This implies that 3628800 = 40320 + 2
They are not same
Hence 10 ! ≠8! + 2!
Statement 10 ! = 8! +2! Is false.
Video Lesson
Video Lesson on Factorials
Related Articles:
Practice Questions on Factorial
- Evaluate 10!/8!
- Is 6! = 3! × 2!
- Simplify (6! × 7!)/3!
- Solve the equation x2 – 36 = 0 and give the answer in factorial
- If (n+1)! = 6n!, solve for n.
