Selection from identical object/article.
The total number of ways of selecting one or more things from ‘p’ identical things of one type, ‘q’ identical things of another type, ‘r’ identical things of another type and ‘n’ distinct things is :
Proof:
Number of ways of selecting 1 thing out of p identical thing = 1
Number of ways of selecting 2 thing out of p identical thing = 1
Number of ways of selecting 3 thing out of p identical thing = 1
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Number of ways of selecting p thing out of p identical thing = 1
Number of ways of selecting 0 thing out of p identical thing = 1
Hence number of ways of selecting 0 or more things out of p identical things = (p + 1)
similarly,
number of ways of selecting 0 or more things out of q identical things = (q + 1)
number of ways of selecting 0 or more things out of r identical things = (r + 1)
Also number of ways of selecting 0 or more out of n different things =
2×2×2×2×………..n times = 2ⁿ
Hence number of ways of selecting 1 or more out of given different things = (p + 1)(q + 1)(r + 1)2ⁿ
Here in above value there is also 1 way where none of the thing is selected.
Hence number of ways of selecting 1 or more of the given things :
Example:- There are 3 identical pens & 4 identical books & 5 identical pencils & 6 different highlighters. In how many ways one or more of these can be selected?
Solution:-
Here p = 3
q = 4
r=5
n = 6
∴ Number of ways = (3 + 1)(4 + 1)(5 + 1)2⁶ – 1
= 7679 Answer