Probability and combinatorics covers selection with and without repeatation,selection with arrangement, derangement, number of integral solutions, grouping, probability, and conditional probability. Check out the questions with detailed solutions given below to reinforce your concepts.

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Instructions

For the following questions answer them individually

Question 1

How many words can be formed from the letters of the word 'ARTISTIC' such that the two 'I's are never together?

Question 2

Given that a>b, how many non negative integral solution exist for the equation a+b+c+d=10?

Question 3

Damani picked up 5 balls at random, one after the other, from a bag full of 150 balls, numbered 1-150. What is the probability that the numbers on the balls, in the order he picks, are in ascending order?

Question 4

If two dice are rolled together, what is the probability that the sum of the dice is less than 7?

Question 5

Amit wants to build a necklace containing 10 different beads marked 1, 2, 3...., 10 such that beads 1, 2 ,3,4,5 are always together and beads 2, 3, 4 are always between 1 and 5. In how many ways can Amit build a necklace.

Question 6

In a bulb factory, three different machines P, Q and R produce 25%, 45% and 30% of the total output. Out of the bulbs produced by P, Q and R 4%, 6% and 5% are found defective. If a defective bulb is picked up randomly, then what is the probability that it is manufactured by P?

Question 7

The number of ways of reaching origin from point (3,3) such that a person can take only one step at a time either along a x axis or along a y axis is

Question 8

The number of ways of reaching origin from point (3,3) such that a person can take only one step at a time either along a x axis or along a y axis is

Question 9

There are โxโ boxes each containing notes of denominations Rs. 10, Rs. 20, Rs. 50, Rs. 100 and Rs. 500 in infinite number. Exactly 9 notes are randomly taken from each of the boxes. What is the minimum possible number of boxes needed to make sure that atleast 8 notes of the same denomination are taken out in total?

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Question 10

Find the number of ways in which 50 perks can be distributed among five people - P, Q, R, S and T - such that Q gets at least 4 perks and T gets a maximum of 7 perks.

Question 11

A whole number is called an even steven number if, when divided by any power of 10, it gives an even quotient. How many even steven numbers are there that are less than 1million?

Question 12

In Telangana, registration numbers for vehicles are made using 3 letters of the English alphabet followed by 3 numbers - 0 through 9. The letters can be in any order but the numbers should be in non-increasing order. A non-increasing order is when the digits may or may not decrease from left to right but they cannot increase. The number of ways of doing this is?

Question 13

On a chess board, all the possible rectangles which are not squares are selected. Find the probability that the rectangle selected has an area of 18 square units with one side as 6 units.

Question 14

A super lotto has all the two digit numbers written on balls kept in a bag. If two balls are chosen at random without replacement what is the probability that their numbers have the same units digit and the product of these numbers has the same units digit?

Question 15

Find the number of ways in which 5 men can watch 4 movies such that each movie is watched by at least one person and no one can watch more than 1 movie.

Question 16

If all the 7 letter permutations of the word 'EXEMPLARY' are ranked in a alphabetical order, what would be the rank of the word 'EXAMPLE'

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Question 17

If X and Y are positive real numbers such that both are less than 4. What is the probability that X+Y > 6?

Question 18

Raghu has to climb 12 steps. He can take 1 step, 2 steps or 3 steps at a time. Find the number of ways in which he can climb the 12 steps.

Question 19

In how many ways can 25 balls be selected from an infinite pool of identical Black, Blue and Burgundy balls?

Question 20

If a man speaks truth 1 out of 4 times and he throws a die and reports a 6, find the probability that the man spoke the truth.

Question 21

In how many ways can we choose two black squares on a chess board so that they do not lie on the same row or column?

Question 22

Two parallel lines have 6 points and 11 points on them. How many different triangles can be formed from using these points as vertices?

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