## Probability Sum - Quants

dharde9
Good Student
Posts: 12
Joined: Fri Aug 21, 2015 4:46 am

### Probability Sum - Quants

An investment guide ranked 20 securities by total returns earned during a certain year. The securities were also labeled as high risk (5 securities), above-average risk (7 securities), average risk (4 securities), below-average risk (2 securities) and low risk (2 securities). In about how many different ways can we take the 20 securities and label 5 of them as high risk, 7 above-average, 4 average risk, 2 below-average risk and 2 low risk, so that each fund is labeled?

edupristine
Finance Junkie
Posts: 709
Joined: Wed Apr 09, 2014 6:28 am

### Re: Probability Sum - Quants

There are total 20 securities.
5 high risk securities will be labeled in: 20C5 ways
Now 15(20 – 5) securities are left. From which 7 above average risk securities has to be labeled in = 15C7 ways
After this, from 8 (5 -7) securities 4 average risk securities has to be labeled in = 8C4 ways
Now, from 4 (8-4) securities 2 below average risk securities has to be labeled in= 4C2 ways
From 2(4-2) securities 2 low risk securities has to be labeled in in= 2C2 ways.
Total no. of ways in which all the securities will be labeled are= 20C5 * 15C7 *8C4 * 4C2 * 2C2
= 15504 * 6435 *70 *6 *1
=41902660800

Calculating this on Texas BA II plus calculator
=> 20 2nd ncr 5 = 15504
=> 15 2nd ncr 7 = 6435
=> 8 2nd ncr 4 = 70
=> 4 2nd ncr 2 = 6
=> 2 2nd ncr 2 = 1
Now multiplying these values = 15504 * 6435 *70 *6 *1
=41902660800

In 41902660800 ways we can label all the securities.