Counting Principles

A bag contains $6$ red and $4$ green balls. A fair die is rolled and a number of balls equal to that appearing on the die is chosen from the bag at random. The probability that all the balls selected are red is

The number of functions $f$ from $A={0,1,2}$ into $B={0,1,2,3,4,5,6,7}$ such that $f(i) \le f(j)$ for $i
1 $8C_3$
2 $8C_3 + 2(8C_2)$
3 $10C_3$
4 None of these
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NIMCET Previous Year PYQ NIMCET NIMCET 2008 PYQ

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Solution

Number of non-decreasing functions = combinations with repetition $= \binom{8+3-1}{3}=\binom{10}{3}$

Twelve villages in a district are divided into 3 zones with 4 villages per zone. The telephone department intends to connect villages so that every two villages in the same zone are connected with three direct lines and every two villages belonging to different zones are connected with two direct lines. How many direct lines are required?

There are 10 points, out of which 6 are collinear. Number of triangles formed:

How many different paths in the $xy$-plane are there from $(1,3)$ to $(5,6)$, if a path proceeds one step at a time either right (R) or upward (U)?

The total number of numbers that can be formed using the digits 3,5 and 7 only if no repetitions are allowed, is.

The number if non –negative integers less than 1000 that contain the digit 1 are.

In how many different ways can the letters of the word “CORPORATION” be arranged so that all the vowels is always come together?

Find the number of elements in the union of 4 sets A, B, C and D having 150, 180, 210 and 240 elements respectively, given that each pair of sets has 15 elements in common. Each triple of sets has 3 elements in common and $A \cap B \cap C \cap D = \phi$

The number of ways in which 5 days can be chosen in each of the 12 months of a non-leap year, is:

The number of bit strings of length 8, that start with the bit 0 or end with the bits 11 is

A professor has 24 text books on computer science and is concerned about their coverage of the topics (P) compilers, (Q) data structures and (R) Operating systems. The following data gives the number of books that contain material on these topics: $n(P) = 8, n(Q) = 13, n(R) = 13, n(P \cap R) = 3, n(P \cap R) = 3, n(Q \cap R) = 3, n(Q \cap R) = 6, n(P \cap Q \cap R) = 2 $ where $n(x)$ is the cardinality of the set $x$. Then the number of text books that have no material on compilers is

The number of one-to-one functions from {1, 2, 3} to {1, 2, 3, 4, 5} is

The number of bit strings of length 10 that contain either five consecutive 0’s or five consecutive 1’s is

Suppose A₁, A₂, ... ₃₀ are thirty sets, each with five elements and B₁, B₂, ...., B_n are n sets each with three elements. Let $\bigcup_{i=1}^{30} A_i= \bigcup_{j=1}^{n} Bj= S$. If each element of S belongs to exactly ten of the Ai' s and exactly nine of the Bj' s then n=

A and B play a game where each is asked to select a number from 1 to 25. If the two numbers match, both win a prize. The probability that they will not win a prize in a single trial is

There is a young boy’s birthday party in which 3 friends have attended. The mother has arranged 10 games where a prize is awarded for a winning game. The prizes are identical. If each of the 4 children receives at least one prize, then how many distributions of prizes are possible?

Naresh has 10 friends, and he wants to invite 6 of them to a party. How many times will 3 particular friends never attend the party?

How many words can be formed starting with letter D taking all letters from the word DELHI so that the letters are not repeated:

Suppose $A_1,A_2,\ldots,A_{30}$ are 30 sets each with five elements and $B_1,B_2,B_3,\ldots,B_n$ are n sets (each with three elements) such that  $\bigcup ^{30}_{i=1}{{A}}_i={{\bigcup }}^n_{j=1}{{B}}_i=S\, $ and each element of S belongs to exactly ten of the $A_i$'s and exactly 9 of the $B^{\prime}_j$'s. Then $n=$

If A={1,2,3,4} and B={3,4,5}, then the number of elements in (A∪B)×(A∩B)×(AΔB)

Lines $L_1, L_2, .., L_10 $are distinct among which the lines $L_2, L_4, L_6, L_8, L_{10}$ are parallel to each other and the lines $L_1, L_3, L_5, L_7, L_9$ pass through a given point C. The number of point of intersection of pairs of lines from the complete set $L_1, L_2, L_3, ..., L_{10}$ is

The number of one - one functions f: {1,2,3} → {a,b,c,d,e} is

The number of all even integers between 99 and 999 which are not multiple of 3 and 5 is

Number of permutations of the letters of the word BANGLORE such that the string ANGLE appears together in all permutations, is