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0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
9! = 362880
10!=3628800

All the numbers after 4! will end with a 0 as we have atleast one "5" in the multiplication.

Hence the units digits will be same after 5! and for 0! and 1!
Now checking the statements

I. n = 10
Yes, the units digits of 10! and 11! are same.
TRUE

II. n = 1
units digit f 1! = 1 and units digit of 2! = 2.
FALSE

III. n = 0
Units digit of 1! = 1 and units digit of 0! = 1.
TRUE

Correct Option: D
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Bunuel
If n! and (n + 1)! have the same units digit, then n can be which of the following?

I. n = 10
II. n = 1
III. n = 0

A. I only
B. II only
C. III only
D. I and III
E. I, II, and III

2 rules to make factorials easy

1. Any factorial greater than ( and including ) 5 will always have units digit as 0
2. 0! = 1! = 1


This question only tests the above two factorial rules...

Answer will definitely be (D)

PS : IMHO this problem can also be solved at a glance if the above properties are known to the test takers.
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Answer: D

(n+1)! = (n+1)n!
1. n=10
Since n! will include multiplication of all numbers from 1 to 10, the unit's digit will be a 0 (any number multiplied by 10 will have unit's digit 0). Similarly, (n+1)! will be multiplication of all numbers from 1 to 11. As this multiplication also includes 10, unit's digit will be 0.
Correct option

2. n=1
n!=1!=1
(n+1)!= 2!= 2
Incorrect option

3. n=0
n! = 0! = 1
(n+1)!= 1!= 1
Correct option
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0! =1

hence I and III
D is correct
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