__All possible selections (dissimilar things)__

In this case there is no bar on ‘r’ term. Out of all
given ‘n’ things, we can select 1 thing or 2 things or 3 things.... up to ‘n’
things at a time.

Number of ways of selecting 1 thing from n things = nC1

Number of ways of selecting 2 things from n things = nC2

and so on and
so on......

Number of ways of selecting n things from n things = nCn

Total number of selections = nC1 + nC2 +
nC3 + .... + nCn

From binomial theorem, nC0 +
nC1 + nC2 + nC3 +
.... + nCn = 2

nC1 + nC2 + nC3 + .... + nCn = 2

^{n}nC1 + nC2 + nC3 + .... + nCn = 2

^{n}- nC0 = 2^{n}– 1**Number of all possible selections from given ‘n’ things = 2**

^{n}– 1

**All possible selections (similar things)**

If p things are
alike of one kind, q things are alike of second kind and r things are alike of
third kind,

then the number of ways of selecting any number of things out of these is

then the number of ways of selecting any number of things out of these is

**(p+1) (q+1) (r+1) – 1**

Explanation:

As similar things exist in each set, we can not simply
apply nPr or nCr.

From the first set of p similar things, all possible
selections are:

selecting nothing, selecting 1 thing, selecting 2 things, etc.,
etc., selecting p things ------ (p+1) ways.

Similarly for the second set -------- (q+1) ways

and for the third set -------- (r+1) ways

Fundamental principle says that, if one work can be
performed in m ways and second work in n ways, then two works can be performed
in m*n ways. Applying that,

the total number of ways becomes: (p+1) (q+1) (r+1)

But this includes a combination of no selections at
all. If we remove that particular combination, the answer becomes: (p+1) (q+1) (r+1) - 1

__Selection of consecutive things:__
Consider the following things which are in a row

A B C D E F G

Select two consecutive things from the above row:

The possibilities are: AB, BC, CD, DE, EF, FG - total 6
ways. Analysing this in reverse-calculation method,

6 = 7 – 2 + 1 = (number of things given) – (number of
things to be selected) + 1 = n- r + 1

**Number of ways of selecting ‘r’ consecutive things from ‘n’ things in a row = n-r+1**

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