BCD adder

Question

Total number of invalid combinations for a BCD adder is

1.   192
2.   100
3.    36
4.   156
   Explanation:
   4-bits are needed to represent a BCD number.
   So, out of 16 (i.e.,2⁴) combinations only 10 are valid. 
   Total combinations=16×16=256
   Valid combinations=10×10=100
   Invalid combinations=256-100=156

I am unable to understand this. Please explain.

Comments

I assume you are familiar with the concept of BCD Adder which takes 2 number to add on a 4-bit Full Adder...
So, 2⁴     
as there are 2 number to be added using BCD adder so total combination (2⁴ * 2⁴ ) = (16 * 16) which is 256 of total combinations.
Now, you must also know that a binary digit can only be accepted by a BCD adder as long as there is no carry out from the MSB  
that is a decimal numbers accepted by a BCD are ( 0 to 9) because when converted to binary, anything greater than 9 will result in a carry out, so we have to manually add (0110)₂ = (6)₁₀
Therefore, there are only 10 (0 to 9) valid combination from both the numbers using BCD adder which is 
10*10=100.
as we have 100 valid combinations, so we rest are automatically invalid. Hence, 256-100 =156 Invalid Combinations.
 

Answer 1

A BCD ranges from 0000 to 1001 (0 to 9) in binary, therefore we need 4 bits to represent BCD.

That means we have a total of 2^4 or 16 combinations of 4 bits available, but we require only 10, i.e.

0000,0001,0010,0011,0100,0101,0110,0111,1000 and 1001

We are left with the remaining 6 or (16-10)  combinations i.e. 1010, 1011,1100, 1101,1110 and 1111

Now, a BCD adder takes 2 two BCD numbers {BCD1, BCD2)

So we have total 16 combinations for each, therefore total combinations we have can have for both BC1 and BCD2 = 16×16=256

Out of 16, only 10 were valid, so if for each 10 were valid inputs for BCD, for both BCD1 and BCD2 we have total 10×10 or 100 valid combination of bits

So invalid inputs that can be given are 256-100 i.e. 156

Comments

Exactly!

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Thank you so much @Abbas2131 @Namit

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If we consider carry-in bit in BCD adder the total combinations will be $2^{9}=512$. The valid input combinations will be $10*10*2=200$ and invalid combinations will be $512-200=312$.

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HI can you please resolve one silly query

0 to 9 is valid and 10 to 15 not valid so 6 combination are not valid

6*6 = 36 are not valid why this is wrong?

Got it

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this is wrong because i am considering only invalid invalid pair we have to also compair the invalid valid pair so.

10 valid * 6 invalid = 60 invalid

6 invalid * 16 valid = 96 invalid

60 +96 = 156 :)