Permutations and Combinations
10.0 Possible Selection from $n$ Identical Objects
10.0 Possible Selection from $n$ Identical Objects
- The number of selections of $r$ objects out of $n$ identical objects is $1$.
- Total number of selections of one or more objects is $n$.
- Total number of selections of none or at least one object is $n+1$.
- Total number of selections of at least one out of $n$, where $p$ are identical, $q$ are identical , $r$ are identical and so on, is given by $$ = [(p + 1)(q + 1)(r + 1)....] - 1$$
Explaination: Let us select atleast one object from $p$, then there are $p$ ways of selecting.
Let us select none from $p$ but atleast one from $q$, then there are $q$ ways of selecting.
Let us select none or atleast one from $p$ and $q$, then there are $(p+1)(q+1)$ ways of selecting.
Extending this to $n$ objects we have, $$ = [(p + 1)(q + 1)(r + 1)....]$$
When none of these are selected, there is one way.
So to select atleast one, there are, $$ = [(p + 1)(q + 1)(r + 1)....] - 1$$
Question 23. There are $5$ tables out of which $2$ are pink, $3$ are blue. Ways of selecting atleast one is?
Solution: Ways of selecting none or atleast one pink, $$\begin{equation} \begin{aligned} = 2 + 1 \\ = 3 \\\end{aligned} \end{equation} $$
Ways of selecting none or atleast one blue, $$\begin{equation} \begin{aligned} = 3 + 1 \\ = 4 \\\end{aligned} \end{equation} $$
Ways of selecting none or atleast one, $$\left( {1,0} \right)\,\left( {1,1} \right)\,\left( {1,2} \right)\,\left( {1,3} \right)\,\left( {2,0} \right)\,\left( {2,1} \right)\left( {2,2} \right)\,\left( {2,3} \right)\,\,\left( {0,1} \right)\,\left( {0,2} \right)\,\left( {0,3} \right)\,\left( {0,0} \right)\, = 12\,$$
Ways of selecting none $=1$
Ways of selecting atleast one = Ways of selecting none or atleast one - ways of selecting none
$$\begin{equation} \begin{aligned} = 12 - 1 \\ = 11 \\ = (2 + 1)(3 + 1) - 1 \\\end{aligned} \end{equation} $$
Question 24. There are $25$ colored balls in a bag. Find the number of ways of selecting if:
(A) All balls are of the same color, and you wish to choose $7$ balls?
(B) All balls are of the same color, and you wish to choose none or more balls?
(C) Six of them are red, four are blue with white stripes, three are yellow, eight are translucent, and the rest are orange and you wish to select atleast one.
Solution: Number of ways of selecting $7$ balls from $25$ identical balls is $25$.
Ways of selecting none or more balls, $$\begin{equation} \begin{aligned} = 25 + 1 \\ = 26 \\\end{aligned} \end{equation} $$
Given that there are 6 red, 4 blue, 3 yellow, 8 translucent ball.
Number of orange balls are, $$\begin{equation} \begin{aligned} = 25 - 6 - 4 - 3 - 8 \\ = 4 \\\end{aligned} \end{equation} $$
Ways of selecting atleast one, $$\begin{equation} \begin{aligned} = [(6 + 1)(4 + 1)(3 + 1)(8 + 1)(4 + 1)] - 1 \\ = [(7)(5)(4)(9)(5)] - 1 \\ = 6300 - 1 \\ = 6299 \\\end{aligned} \end{equation} $$
