How many 4-digit integers have the form abcd, where b is even and d =

Question

How many 4-digit positive integers have the form abcd, where b is even and d >= 2b?

A. 900 
B. 1200 
C. 1620 
D. 2400 
E. 2700

PiyushYadav1995 · Answer

Here
Abcd B is even and d>=2b
Let b =0 then D>=0 total 0-9 (10 ways)
A can take values from 1-9 as the question says 4 digit numbers
B is 0 1 value
C has 10 ways
D has 10 ways 
ABCD can be written with b 0 in 9*1*10*10 = 900 ways

Similarly
B=2 
A 9 ways
B 1 way
C 10 ways
D 6 ways (4-9 ,the values d can take )
total = 9*1*10*6=540 ways

B=4 
D has 2 ways
A 9 ways 
B 1way 
C 10 ways
total=9*1*10*2=180 ways

Total number ways = 900+540+180=1620 ways.

How can answer be C????

MathRevolution . Please Comment.

Please give kudos if i am correct.

SSM700plus · Answer

Hello,

Case 1 :-   
Consider b =0: 
a can take 9 values from 1 to 9. - 9
b should be even - only 1 value - 1
c - any of the 10 values can be selected - 10.
d >=2b. It implies d can take any value >=0. - 10 values.

Sum = 9*1*10*10 = 900.

Case 2 :-   
Consider b =2: 
a can take 9 values from 1 to 9. - 9
b should be even - only 1 value - 1
c - any of the 10 values can be selected - 10.
d >=2b. It implies d can take any value >=4. - 6 values.

Sum = 9*1*10*6= 540.

Case 3 :-   
Consider b =4: 
a can take 9 values from 1 to 9. - 9
b should be even - only 1 value - 1
c - any of the 10 values can be selected - 10.
d >=2b. It implies d can take any value >=8. - 2 values.

Sum = 9*1*10*2= 180.

Adding all the cases :- 900 + 540 + 180 = 1620. ( Answer )

Please let me know where I am going wrong. I feel there is typo in Answer Choice.

PrabhaMithila · Answer

I am also getting 1620 as the answer, instead of 1520

MathRevolution · Answer

=>

Suppose abcd is a 4-digit number.
There are 9 possible values for a: a = 1, 2, …, 9.
There are 10 possible values of c: c = 0,1,2,…,9. 
Since b is even, b can take on the values 0,2,4,6 and 8.
However, the condition d >= 2b limits the possible values of b to 0, 2 and 4.

Case 1 : b = 0 => d = 0, 1, … , 9
The number of possible values of d is 10.
There are 10 * 9 * 10 = 900 4-digit integers with b = 0.

Case 2: b = 2 => d = 4, 5, …, 9
The number of possible values of d is 6.
There are 6 * 9 * 10 = 540 4-digit integers with b = 2.

Case 3: b = 4 => d = 8, 9
The number of possible values of d is 2.
There are 2 * 9 * 10 = 180 4-digit integers with b = 4.
In total, there are 900 + 540 + 180 = 1620 possible 4-digit integers of this form.

Therefore, the answer is C.

Answer: C

SSM700plus · Answer

900 + 540 + 180 is not 1520. It adds up to 1620. Please correct the answer choices. It should be 1620

PiyushYadav1995 · Answer

Hello  MathRevolution

I am sorry. But there is a calculation mistake in your solution. 900 + 540 + 180 = 1520? This should be 1620. Please correct.

mikemcgarry   souvik101990   chetan2u . Please correct me if i am wrong.

Please give me KUDOS

HisHo · Answer

Hi  MathRevolution

Why you didn't take the negative integers into consideration? Remember that the Q didn't put a restriction to be +ve numbers.
In this case, we should have 1620 x 2 =3240. Am I right?

MathRevolution · Answer

Hi, 
Since the 4-digit integers have the form "abcd" and "abcd" doesn't have a sign, we can presume those integers are positive.
Anyhow, the question has been updated for more clear expression.

Thank you for your comments.

PKN · Answer

Kindly refer the enclosed grid for the solution.

From the enclosure, the required no of 4-digits nos is found to be 1620.

correct answer option (C)

GyMrAT · Answer

Given the 4 digit number of the abcd, with b- even & d>=2b

a - can take values from 1-9 (except 0) = 9 values
b - can take single digit even # - 0,2,4,6,8 = 5 values
c - can take values from 0-9 = 10 values

Now for d>=2b , when 
b = 0, d>=0 can take values from 0-9 = 10 values
b = 2, d>=4 can takes values as 4,5,6,7,8,9 = 6 values
b = 4, d>=8 can take values as 8,9 = 2 values

Hence total#'s formed = (9*1*10*10)+(9*1*10*6)+(9*1*10*2) = 900+540+180 = 1620

Answer C.

Thanks,
GyM

exc4libur · Answer

A={1 to 9}=9 options
C={0 to 9}=10 options

9≥D≥2B
8≥B=Even≥0
B={0,2,4,6,8}
D≥2B={0,4,8,12,16} 
but D≤9…2B={0,4,8} and B={0,2,4}

B=0, then D≥2B={0 to 9}=10 options… ABCD=(9)(1)(10)(10)=900
  B=2, then D≥2B={4 to 9}=6 options… ABCD=(9)(1)(10)(6)=540
  B=4, then D≥2B={8 to 9}=2 options… ABCD=(9)(1)(10)(2)=180

total: =900+540+180=1620

Answer (C)

satya2029 · Answer

b is even means b can be 0,2,4,6,8 
so 2b can be 0,4,8,12,16
 d>=2b 
 start with 0

abcd
for a= 9 options, b= 1 option, c= 10 options, d= 10 options
Total=9*1*10*10=900

Then b=2, 2b=4 so d=4,5,6,7,8,9

a= 9 options, b= 1 option, c=10 options, d=6
Total=9*1*10*6=540

then b=4, 2b=8 so d=8,9 
a= 9 options, b =1 option, c=10 options, d= 2 options
Total=180
So total=900+540+180
1620
C:)

bumpbot · Answer

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How many 4-digit integers have the form abcd, where b is even and d >=

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