Problem Solving

How many integers between 0 and 1560 have a prime tens digit and a prime units digit.

A) 295
B) 252 OA
C) 236
D) 96
E) 76

I was able to solve this question by two methods.

First is through permutations, I am getting 232 as the answer. Took me less than 1 minute to solve this way.

Second, finding the pattern in counting the numbers from 1560. But that is very time consuming, took me around 3-4 minutes.

While posting the answers, kindly post the time it took you to solve and the explanation.

Thanks.

Well, it depends on how you work, I chose the pattern and it was pretty fast, less 1', and with less error probability in my case. The only point is that I took 1 into account so it was wrong but the pattern was ok. The answer was noth there so it took me more time to correct it. Hope it helps even if I was wrong.

Isn't 252 the right answer

between 1- 100 = 16 numbers
between 100 - 1000 = 16 * 9 = 144 numbers
between 1000 - 1500 = 80 numbers
between 1500 - 1560 = 12 numbers

16 + 144 + 80 + 12 = 252

Am I doing something wrong?

How would you do this using permutations?

-A

No you are not doing anything wrong, your answer is perfect.
I also did it by the same method but took me more than 3 minutes.

Now permutation method.

0,1,2,3,4,5,6,7,8,9

We have 4 primes numbers (2,3,5,7)

For two digits number 4*4 = 16

For three digits number 9*4*4 =144

For Four digits the number cannot exceed 1560
Therefore, 1*6*3*4 = 72
1 because first digit cannot have more than 1 options (1)
6 because second digit cannot have more than 6 options (0,1,2,3,4,5)
3 because third digit cannot have more than 3 prime number (2,3,5)
4 because we can have all the prime numbers.

16+144+72=232

Let me know if I am doing something wrong.

Thanks,

I have done this way:

Since we need a only tens and units digit to be prime, we just need combinations of 2,3,5 and 7 starting from 10 till 1560.So it is pretty much clear that for every 100 numbers (starting from 0 to 100) we will have same number of combinations.So for first 100 numbers(0 to 100) we have 2^4 combinations i.e 16.

for all 15 such calculations(till 1500) we have 15*16 = 240 combinations.
In last 60 we have 12 prime number combnations.

Total = 240 + 12 = 254

or ?

ricky wrote:I have done this way:

Since we need a only tens and units digit to be prime, we just need combinations of 2,3,5 and 7 starting from 10 till 1560.So it is pretty much clear that for every 100 numbers (starting from 0 to 100) we will have same number of combinations.So for first 100 numbers(0 to 100) we have 2^4 combinations i.e 16.

for all 15 such calculations(till 1500) we have 15*16 = 240 combinations.
In last 60 we have 12 prime number combnations.

Total = 240 + 12 = 254

or ?

Nicely Done!!!

parallel_chase wrote:No you are not doing anything wrong, your answer is perfect.
I also did it by the same method but took me more than 3 minutes.

Now permutation method.

0,1,2,3,4,5,6,7,8,9

We have 4 primes numbers (2,3,5,7)

For two digits number 4*4 = 16

For three digits number 9*4*4 =144

For Four digits the number cannot exceed 1560
Therefore, 1*6*3*4 = 72
1 because first digit cannot have more than 1 options (1)
6 because second digit cannot have more than 6 options (0,1,2,3,4,5)
3 because third digit cannot have more than 3 prime number (2,3,5)
4 because we can have all the prime numbers.

16+144+72=232

Let me know if I am doing something wrong.

Thanks,

You calculated

For Four digits the number cannot exceed 1560
Therefore, 1*6*3*4 = 72

That 3 in ur expression coreesponds to digits 2,3 and 5....means 10's digits can only be 2,3 or 5. However it is true only for numbers of form 15xx and not for others of the form 10xx, 11xx, 12xx, 13xx and 14xx. Hence u gettin the ans wrong.

u shud solve like
For Four digits the number cannot exceed 1560
Therefore, 1*6*4*4 - 1*1*1*4 = 92

acecoolan wrote:Isn't 252 the right answer

between 1- 100 = 16 numbers
between 100 - 1000 = 16 * 9 = 144 numbers
between 1000 - 1500 = 80 numbers
between 1500 - 1560 = 12 numbers

16 + 144 + 80 + 12 = 252

Am I doing something wrong?

How would you do this using permutations?

-A

Yours is the faster I think. Here what I did.

1-100: 16 numbers
So 15*16=240
1500-1560: 12
So 252