Question

If two states are selected at random from a group of 20 ​states, determine the number of possible outcomes if the group of states are selected with replacement or without replacement.

Answer 1

Answer:

With replacement

 21C2 = 210 outcomes

without replacement

20C2 = 190 outcomes

Step-by-step explanation:

For determining the number of possible outcomes you need count the number of possible combinations, because a combination is a selection of a number of items from a set of items where the order of selection does not matter.

The number of possible combinations is calculated thus

nCr = $\frac{n!}{(n-r)!r!}$

     

           Where n: number of items of the set

                        r: number of selected items

a) If the group of states are selected with replacement then

   (n+r-1)Cr  

   n = 20 states

   r = 2 states

  then n +r -1 = 20 +2 -1 = 21

  21C2 = $\frac{21!}{(21-2)!2!} = 210$

b) If the group of states are selected without replacement then

 nCr

 n = 20

 r = 2

20C2 = $\frac{20!}{(20-2)!2!} = 190$

   

Answer 2

The selection of the two states is an illustration of combination

• There are 210 ways to select 2 states from 20 with replacement
• There are 190 ways to select 2 states from 20 without replacement

The given parameters are:

$\mathbf{n = 20}$ --- total number of states

$\mathbf{r = 2}$ -- selected states

(a) When the two states are selected with replacement

To do this, we make use of the following combination formula

$\mathbf{^{n+r-1}C_r = \frac{(n + r - 1)!}{(n - 1)!r!}}$

So, we have:

$\mathbf{^{20+2-1}C_2 = \frac{(20 + 2 - 1)!}{(20 - 1)!2!}}$

Simplify

$\mathbf{^{21}C_2 = \frac{21!}{19!2!}}$

Expand

$\mathbf{^{21}C_r = \frac{21 \times 20 \times 19!}{19! \times 2 \times 1}}$

$\mathbf{^{21}C_r = \frac{21 \times 20}{2}}$

$\mathbf{^{21}C_r = 21 \times 10}$

$\mathbf{^{21}C_r = 210}$

Hence, there are 210 ways to select 2 states from 20 with replacement

(b) When the two states are selected without replacement

To do this, we make use of the following combination formula

$\mathbf{^{n}C_r = \frac{n!}{(n - r)!r!}}$

So, we have:

$\mathbf{^{20}C_2 = \frac{20!}{(20 - 2)!2!}}$

Simplify

$\mathbf{^{20}C_2 = \frac{20!}{18!2!}}$

Expand

$\mathbf{^{20}C_2 = \frac{20 \times 19 \times 18!}{18! \times 2 \times 1}}$

$\mathbf{^{20}C_2 = \frac{20 \times 19}{2}}$

$\mathbf{^{20}C_2 = 10 \times 19}$

$\mathbf{^{20}C_2 = 190}$

Hence, there are 190 ways to select 2 states from 20 without replacement

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