GATE Electrical 2018 | GA | Question: 6

Question

An e-mail password must contain three characters. The password has to contain one numeral from $0$ to $9,$ one upper case and one lower case character from the English alphabet. How many distinct passwords are possible$?$

• A. $6,760$
• B. $13,520$
• C. $40,560$
• D. $1,05,456$

Comments

Each combination can occur 3! times.. right?

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Oh! yes you are right.

Answer 1

Select any digit from $10$ digits, one uppercase alphabet from $26$ alphabets and  one lowercase  alphabets  from $26$ alphabet, each combination can permute $3!$ times for ex: $5Ab, 5bA, A5b, Ab5, bA5, b5A$

So, total possible passwords= $(10C1 * 26C1 * 26C1 ) * 3! $ 

$= 40560$

Hence option c) is correct

Comments

(10C1*26C1*26C1)*6

This is a Correct Formula

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$^{10}C_1 \times ^{26}C_1 \times ^{26}C_1 \times 3!$

$\implies ^{10}C_1 \times ^{26}C_1 \times^ {26}C_1 \times 3 \times 2$

$\implies ^{10}C_1 \times ^{26}C_1 \times ^{26}C_1 \times 6$

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Some extra info for nerds.

After getting  10 * 26 * 26 * 3 * 2. We know that the final answer is a multiple of 10, so it will have 0 as the last position hence option D is eliminated, rest of the options have 0 as the last position.

The final answer is also a multiple of 3. Among A, B and C only option C is multiple of 3. Hence option C is the answer.

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Elimination method.

Answer 2

Here password is consist of 3 characters; 

_ _ _ 

1. First dash can be fill with 0 to 9 numbers: so total possibilities are 10.
2.  Second dash can be fill with 26 letters which are UPPERCASE : Total possibilities are 26
3. Third dash can be fill with 26 small letters: total possibilities are 26

thus total no of distinct password are : 26*26*10 = 6760 

but take an eg, a A 0 it can be arrange in 6 different ways 

1. a 0 A
2. a A 0
3. 0 a A 
4. 0 A a
5. A a 0
6. A 0 a

thus total possible arrangement can be 6760*6 = 40560 (distinct possible ways for a password).