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INTRODUCTION OF COMBINATIONS is considered one the most difficult concept.
200 Questions around this concept.
The value of is equal to:
n is selected from the set {1,2,3......49} and the number 2n+3n+5n is formed. Total number of ways of selecting n so that the formed number is divisible by 4 is equal to
If and
, determine the values of n and r.
${ }^{n-1} C_r=\left(k^2-8\right)^n C_{r+1}$ if and only if:
Let $\alpha=\frac{(4 !) !}{(4 !)^{3 !}}$ and $\beta=\frac{(5 !) !}{(5 !)^{4 !}}$ Then :
The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is
A bag contains 4 red balls, 3 green balls, and 2 blue balls. If you randomly select 3 balls from the bag without replacement, what is the number of possible outcomes?
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A bag contains 4 red marbles, 3 blue marbles, and 2 green marbles. If you randomly select 2 marbles from the bag without replacement, what is the number of possible outcomes?
A bakery offers 4 types of cupcakes: chocolate, vanilla, strawberry, and lemon. If a customer wants to choose 2 cupcakes for a special offer, how many different combinations of cupcakes can they select?
A box contains 3 white, 4 black, and 5 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?
So far our task was always to “arrange” objects i.e. to place them in a specific order among themselves.
Sometimes we would be interested in only “selecting” a few objects out of the given objects. In this case, we just need to “select” and we do not need to “arrange” them in an order.
The notation of selecting r objects from n given object is ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}$.
${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} = \frac{n!}{(n-r)!\cdot r!}$
Multiplication Rule
If a certain work W can be completed by doing 2 tasks, first doing task A AND then doing task B. A can be done in m ways and following that B can be done in n ways, then the number of ways of doing the work W is (m x n) ways.
Addition Rule
If work W can be completed by doing task A OR task B, and A can be done in m ways and B can be done in n ways (and both cannot occur simultaneously: in this case we call tasks A and B as mutually exclusive), then work W can be done in (m + n) ways.
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