Question

There should not be a "disconnect between your work and the answer Lists Exercise 8.6 How many 3-element lists can be formed whose entries are drawn from a pool of n elements - if the first and last entries of the list must be the same? Ans.n? a) b) if the first and last entries of the list must be different?

Answer 1

Answer:

(a) The no. of 3 elements list is $n^{2}$

(b) The no. of 3 elements list is $n^{3} - n^{2}$

Step-by-step explanation:

As per the question:

Total no. of possible elements = n

The no. of elements required out of n elements to form the list = 3

Now,

(a) When the first and last entries on the list are same:

No. of ways for the entry of first element = n ways

No. of ways for the entry of second element when it can be entered without any restriction = n ways

No. of ways for the entry of last element as it has to be the same as the first element = 1 way  

Thus the total no. of 3 elements list according to the multiplication principle:

$n\times n\times 1 = n^{2}$

(b) When the first and the last entries must be distinct:

No. of ways for the entry of first element = n ways

No. of ways for the entry of second element when it can be entered without any restriction = n ways

No. of ways for the entry of last element as it has to be different from the first element = (n - 1) ways

Thus the total no. of 3 elements list according to the multiplication principle:

$n\times n\times (n - 1) = n^{2}(n - 1)\ ways = n^{3} - n^{2}$