Question

15.

a. Using the formula for combinations, show that the number of ways of selecting 2 items from a group of 3 items is the same as the number of ways to select 1 item from a group of 3.
b. Show that n???????? and n????n−???? are equal. Explain why this makes sense.

Answer 1

a. selecting 2 from 3 using combination formula= 3way

b. selecting 1 from 3 using combination= 3way

Step-by-step explanation:

using combination formula that is:

3C2 = 3?÷(3-2)?2?= 3ways

3C1 = 3?÷(3-1)?1? = 3ways

b. n?= nx(n-1)(n-2)(n-3)(n-4)-------

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Answer 2

Step-by-step explanation:

a)

`{3 \choose 2} = (3!)/(2!(3-2)!) = (6)/(2*1) = 3`

`{3 \choose 1} = (3!)/(1!(3-1)!) = (6)/(1*2) = 3`

b) First, we have that

`{n \choose k} = (n!)/(k!(n-k)!)`

On the other hand,

`{n \choose n-k} = (n!)/((n-k)!(n(n-k))!) = (n!)/((n-k)!k!)`

Therefore, both expressions are equal. This makes sense, because selecting k elements from a group of n is the same than specify which elements you will not select. In order to specify those elements you need to select the n-k elements that will not be selected. Hence, each time you are selecting k from n, you are also selecting n-k from n, and from that reason both combinatorial numbers are equal.