# What is the difference between Selections and Arrangements/Permutaions?

• Nov 5th 2009, 08:47 PM
22upon7
Probability Application help
Hi,

I'm revising for my Maths exam on Tuesday and I really need help with this question:

Three letters are chosen at random from the word "HEART" and arranged in a row. Find the probability that:

a. the letter H is first
b. the letter H is chosen
c. both vowels are chosen

I think this is how you work a. out:

HEART has 5 letters, so 1/5 = 0.2

I'm not sure how to work b. and c. out though, I think it needs permutation and/or combinations but I can't work out the correct answer.

Any help would be greatly appreciated, and I'm sure I'll have more questions to come.

Thanks,

Dru
• Nov 5th 2009, 10:10 PM
Robb
Quote:

Three letters are chosen at random from the word "HEART" and arranged in a row. Find the probability that:

a. the letter H is first

So, three letters are chosen randomly and arranged in a row. There are $\displaystyle 5P3=60$ ways they can be arranged... Since you want the first letter to be a H, there is only one way that can happen, and there is $\displaystyle 4P2=12$ ways that can happen. So $\displaystyle Pr(\mbox{H is first})=\frac{1\cdot 12}{60}=\frac{1}{5}$

Quote:

b. the letter H is chosen
Letter H can now be in 1 of 3 places, so
$\displaystyle P(\mbox{Letter H is chosen})=\frac{3\cdot 12}{60}=\frac{3}{5}$

Quote:

c. both vowels are chosen
So there is $\displaystyle 2P2=2$ ways to arrange the 2 vowels, and the remaining 3 characters can be arranged in any of the 3 positions, so there are 9 ways for the other letter to be positioned.
So $\displaystyle P(\mbox{both vowels are chosen})=\frac{2\cdot 9}{60}=\frac{3}{10}$
• Nov 5th 2009, 10:49 PM
22upon7
Quote:

Originally Posted by Robb
So, three letters are chosen randomly and arranged in a row. There are $\displaystyle 5P3=60$ ways they can be arranged... Since you want the first letter to be a H, there is only one way that can happen, and there is $\displaystyle 4P2=12$ ways that can happen. So $\displaystyle Pr(\mbox{H is first})=\frac{1\cdot 12}{60}=\frac{1}{5}$

Thanks Robb, I didn't understand this bit though, could you help me out with it?:

Quote:

Originally Posted by Robb
Since you want the first letter to be a H, there is only one way that can happen, and there is $\displaystyle 4P2=12$ ways that can happen.

Thanks again,

Dru
• Nov 5th 2009, 11:02 PM
Robb
Just the permuation formula,
$\displaystyle \frac{4!}{(4-2)!}=\frac{24}{2}=12$
• Nov 5th 2009, 11:28 PM
22upon7
yes, but how did you get those values? The 4 and the 2?

Thanks again,

Dru
• Nov 5th 2009, 11:48 PM
Robb
Oh, sorry. Orignally you had 5 letters to choose from, and you were selecting 3. But because the first one is a H, you are now selecting 2 letters from the remaining 4.
So the probability is just the ways you can arrange H with 2 other letters, divided by the total number of ways you could arrange 3 letters from 5
• Nov 6th 2009, 12:17 AM
22upon7
No worries, Thanks again mate!