The question is from Permutation and Combination. This question combines number theory with combinatorics. We need to find out the number of multiples of 12 which comprises of only 2's and 3's. This section hosts a number of questions which are on par with CAT questions in difficulty on CAT Permutation and Combination, and CAT Probability.

Question 5: A seven-digit number comprises of only 2's and 3's. How many of these are multiples of 12?

- 11
- 12
- 10
- 22

11

Try upto 40 hours for free

Learn from the best!

Limited Seats Available - Register Now!

Number should be a multiple of 3 and 4. So, the sum of the digits should be a multiple of 3. We can either have all seven digits as 3, or have three 2's and four 3's, or six 2's and a 3.

(The number of 2's should be a multiple of 3).

For the number to be a multiple of 4, the last 2 digits should be 32. Now, let us combine these two.

All seven 3's - No possibility.

Three 2's and four 3's - The first 5 digits should have two 2's and three 3's in some order.

No of possibilities = \\frac{5!}{3!2!}\\) = 10

Six 2's and one 3 - The first 5 digits should all be 2's. So, there is only one number 2222232.

So, there are a total of 10 + 1 = 11 solutions.

The question is **"A seven-digit number comprises of only 2's and 3's. How many of these are multiples of 12?"**

Choice A is the correct answer.

1. Digit Numbers | CAT Permutation and Combination | CAT Coaching Online

2. Numbers with Distinct Digits | CAT Permutation and Combination | CAT Coaching Online

3. Alphabetical Order | CAT Permutation and Combination | CAT Coaching Online

4. Geometric Progression | CAT Permutation and Combination | CAT Coaching Online

5. Bayes Theorem | CAT Permutation and Combination | CAT Coaching Online

2. Numbers with Distinct Digits | CAT Permutation and Combination | CAT Coaching Online

3. Alphabetical Order | CAT Permutation and Combination | CAT Coaching Online

4. Geometric Progression | CAT Permutation and Combination | CAT Coaching Online

5. Bayes Theorem | CAT Permutation and Combination | CAT Coaching Online

6. Rearranging Letters | CAT Permutation and Combination | CAT Coaching Online

7. Possible Integer Solutions | CAT Permutation and Combination | CAT Coaching Online

8. Rearrangement of Letters | CAT Permutation and Combination | CAT Coaching Online

9. Number Theory | CAT Permutation and Combination | CAT Coaching Online

10. Distribution | CAT Permutation and Combination | CAT Coaching Online

7. Possible Integer Solutions | CAT Permutation and Combination | CAT Coaching Online

8. Rearrangement of Letters | CAT Permutation and Combination | CAT Coaching Online

9. Number Theory | CAT Permutation and Combination | CAT Coaching Online

10. Distribution | CAT Permutation and Combination | CAT Coaching Online

Copyrights © All Rights Reserved by 2IIM.com - A Fermat Education Initiative.

Privacy Policy | Terms & Conditions

CAT^{®} (Common Admission Test) is a registered trademark of the Indian Institutes of Management. This website is not endorsed or approved by IIMs.

2IIM Online CAT Coaching

A Fermat Education Initiative,

58/16, Indira Gandhi Street,

Kaveri Rangan Nagar, Saligramam, Chennai 600 093

**Phone:** (91) 44 4505 8484

**Mobile:** (91) 99626 48484

**WhatsApp:** WhatsApp Now

**Email: **prep@2iim.com