# CAT Quantitative Aptitude Questions | CAT Permutation and Combination Questions

###### CAT Questions | Permutation Probability | Counting 7 Digit Numbers CAT Questions

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?

1. 11
2. 12
3. 10
4. 22

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##### Method of solving this CAT Question from Permutation and Combination: One unit number theory and one unit Combinatorics makes a potent combination for fabulous questions!

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?" ##### Hence the answer is "11" Choice A is the correct answer. ###### Best CAT Online Coaching Try upto 40 hours for free Learn from the best! ###### Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2021Enroll at 49,000/- 44,000/- Online Classroom Batches Starting Now! ###### Best CAT Coaching in ChennaiPrices slashed by Rs 4000/- Attend a Demo Class ## CAT Preparation Online | CAT Permutation and Combination; Probability Videos On YouTube #### Other useful sources for Permutation and Combination Questions | Probability Questions ##### Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 ##### How to reach 2IIM? Phone:$91) 44 4505 8484
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