Question

A question paper is divided into two parts $\mathrm{A}$ and $\mathrm{B}$ and each part contains 5 questions. The number of ways in which a candidate can answer 6 questions selecting at least two questions from each part is:
A.80
B.100
C.200
D.None of these

Answer 1

Hint: Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The probability formula is used to compute the probability of an event to occur. To recall, the likelihood of an event happening is called probability.
- Probability Rule One (For any event $A, 0 \leq P(A) \leq 1)$
- Probability Rule Two (The sum of the probabilities of all possible outcomes is 1 )
- Probability Rule Three (The Complement Rule)
- Probabilities Involving Multiple Events.
- Probability Rule Four (Addition Rule for Disjoint Events)
- Finding $\mathrm{P}(\mathrm{A}$ and $\mathrm{B})$ using Logic.

Complete step-by-step answer:
Permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor.
The formula for permutations is: $\mathrm{^nP_r}=\dfrac {\mathrm{n!}} {(\mathrm{n}-\mathrm{r}) !}$
The formula for combinations is: $\mathrm{^nC_r}=\dfrac {\mathrm{n!}} {[\mathrm{r} !(\mathrm{n}-\mathrm{r}) !]}$
A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, we can select the items in any order. Combinations can be confused with permutations. In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.
A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, we can select the items in any order. Combinations can be confused with permutations. In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.

A	B	Total
5 questions	5 questions
2	4	6
3	3	6
4	2	6

There are two section and the candidates has to answer at least
2 questions from each section with a total of 6 questions.
So, there are three following ways.
(i) no of ways $\Rightarrow$ selecting 2 questions from section $\mathrm{A}$ and 4 questions from sections $\mathrm{B}$ $\Rightarrow^{5} \mathrm{C}_{2} \times{ }^{5} \mathrm{C}_{4} \Rightarrow 10 \times 5 \Rightarrow 50$
(ii) no. of ways $\Rightarrow$ selecting 3 questions from each section. $\Rightarrow{ }^{5} \mathrm{C}_{3} \times{ }^{5} \mathrm{C}_{3} \Rightarrow 10 \times 19=100$
(iii) no. of ways $\Rightarrow$ selecting 4 from section $\mathrm{A}$ and 2 from section$\mathrm{B}$
$\Rightarrow^{5} \mathrm{C}_{4} \times{ }^{5} \mathrm{C}_{2} \Rightarrow 5 \times 10 \Rightarrow 50$
$\therefore$ Total no. of ways $=50+100+50=200$.
So, the correct answer is option C.

Note: One could say that a permutation is an ordered combination. The number of permutations of $\mathrm{n}$ objects taken $\mathrm{r}$ at a time is determined by the following formula:
$\mathrm{P}(\mathrm{n}, \mathrm{r})=\dfrac {\mathrm{n} !}{(\mathrm{n}-\mathrm{r}) ! \mathrm{n} !}$ is read $\mathrm{n}$ factorial and means all numbers from 1 to $\mathrm{n}$ multiplied. Combinations are a way to calculate the total outcomes of an event where order of
the outcomes do not matter.
To calculate combinations, we will use the formula $\mathrm{^nC_r}=\dfrac {\mathrm{n!}} {\mathrm{r!}^{*}(\mathrm{n}-\mathrm{r}) !}$ where $\mathrm{n}$ represents the total number of items, and $\mathrm{r}$ represents the number of items being chosen at a time. Thus, $\operatorname{^nP_r}(\mathrm{n}, \mathrm{r})$ The number of possibilities for choosing an ordered set of $\mathrm{r}$ objects $(\mathrm{a}$ permutation) from a total of n objects. Definition: $\operatorname{^nP_r}(\mathrm{n}, \mathrm{r})=\dfrac {\mathrm{n} ! }{(\mathrm{n}-\mathrm{r}) ! \mathrm{^nC_r}(\mathrm{n}, \mathrm{r})}$.