Suppose that for an experimental device setup we have to choose settings for three parameters. There are 4, 3 and 5 settings for parameters 1, 2 and 3 respectively. The order in which the settings will be chosen is not does not matter. How may configurations does the device allow

Number of configurations does the device allow = 1

Explanation:

Given:

Number of parameters = 3

setting to choose for parameters = 3

Order does not matter

To find:

Number of configurations the device allows

Solution:

We will use Combinations here. This cannot be achieved through permutations because we have to choose settings 1,2,3 (3 settings) for three parameters 4,3,5 and the order does not matter but in permutation order does matter. So here we use Combinations. The formula for combination is:

C(n,r) = n! / r! (n-r)!

Here the number of parameters is 3 i.e. 1,2,3 and settings are also 3 i.e. 4,3,5 So,

n = 3

r = 3  

Putting these values in the above formula we get:

nCr  = n! / r! (n-r)! = 3! / 3! (3-3)!

                            =  3*2*1 /3*2*1 (0)!              

                            =   3*2*1 /3*2*1 (1)               Since 0! = 1

                            = 6 / 6

                nCr      = 1

So when order of settings 4,3,5 does not matter the number of configurations that device can allow is 1.

If the repetition is allowed and order does not matter then we can use the following formula:

(r + n - 1)! / r!(n-1)! = (3 + 3 - 1)! / 3! (3-1)!

                           = 5! / 3! 2!

                           = 5*4*3*2*1 / (3*2*1)(2*1)

                           = 120 / 12

(r + n - 1)! / r!(n-1)! = 10

For example if we have same setting for more than 1 parameters the combinations can be:

333, 444, 555, 445, 443, 554, 553, 334, 335, 345

But it is unlikely that a setting can be repeated for more than 1 parameter a a time.

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