Combinatorics #5

[papgust]

From a set of three capital consonants, five small consonants and 4 small vowels, how many words can be made each starting with a capital consonant and containing 3 small consonants and two small vowels?

A. 3600
B. 7200
C. 21,600
D. 28,800

Solution and explanation pls?

[NikolayZ]

Papgust, are you sure you posted answer choices right? =)

This is a permutation problem, as order of letters matters.
So, there are 3 possibilities for capital consonant in a word.

5!/2! - small consonants = 60

and 4!/2! - small vowels = 12

Total quantity of words will be 3*60*12=2160

[papgust]

NikolayZ,

I took this problem from Arun sharma's Quantitative aptitude for CAT. I believe that the answer choices must be correct.

[Stuart@KaplanGMAT]

NikolayZ,

I took this problem from Arun sharma's Quantitative aptitude for CAT. I believe that the answer choices must be correct.

I've never heard of the source, but here are two problems with the question:

1) there are only 4 answer choices, so it's clearly not designed for the GMAT; and

2) it's unanswerable without further information. In particular, we need to know if the letters in each category are distinct.

For example, if the 3 capital consonants were A, A and A, we'd only have 1 choice for our first letter. If the 3 capital consonants were A, B and C, we'd have 3 such choices.

If we assume that all the letters in each category are distinct (something we'd never have to do on a more realistic question):

Part 1: 3 choices for the first letter.

Part 2: first, let's select our potential letters.

Consonants - choosing 3 out of 5 = 5C3 = 5!/3!2! = 5*4/2 = 10

Vowels - choosing 2 out of 4 = 4C2 = 4!/2!2! = 4*3/2 = 6

So, we have 10*6 = 60 possible selections of letters.

Second, let's see in how many different ways we can arrange those 5 letters. Simple enough, there are n! different ways to arrange n distinct objects, so 5!.

Therefore, our final answer is:

3*60*5! = 180 * 120 = 21600

However, as I said above, this is a very poorly written question and, as a result, I'd be suspicious of anything from that source.

[papgust]

Thanks Stuart!

I missed the step of multiplying by 5! for rearranging those 5 letters. Thanks for pointing it out.

The source is actually not well-known globally but it's good recommended book in india for the preparation of CAT exam. I'm using this source just to strengthen my basics.