# nCr Definitions and Properties

## Trending Questions

**Q.**

Given $n\left(U\right)=20,n\left(A\right)=12,n\left(B\right)=9,n(A\cap B)=4$, where $\cup $ is the universal set, $A$ and $B$ are subset of $\cup $, then $n(A\cup {B}^{c})=$

$17$

$9$

$11$

$3$

$16$

**Q.**

If $n\left(A\right)=3,n\left(B\right)=5$and $n(A\cap B)=2$ then $n\left[\right(A\times B)\cap (B\times A\left)\right]=$

$5$

$3$

$4$

$6$

**Q.**

In how many ways can a committee be formed of 5 members from 6 men and 4 women if the committee has at least one woman

- 186
- 252
- None of these
- 246

**Q.**The number of ways in which 5 different balls can be placed in 3 identical boxes such that no box remains empty is

**Q.**If 683+883 is divided by 49, then the remainder is

- 35
- 5
- 1
- 0

**Q.**

if $A=\left\{{4}^{n}-3n-1:nbelongstoN\right\}$ and $B=\left\{9(n-1):nbelongstoN\right\}$, then

$B\subset A$

$A\cup B=N$

$A\subset B$

None of these

**Q.**

**If polynomials **$a{x}^{3}+3{x}^{2}-3$** and **$2{x}^{3}-5x+a$** when divided by **$(x-4)$** leave the remainders as **${R}_{1}$** and **${R}_{2}$** respectively. **

**Find the values of **$a$** in **$2{R}_{1}-{R}_{2}=0$

**Q.**If A =[i 0] B=[0 i] , where i=square root of [0 -i] [i 0] Minus one, then the correct relation is 1.A+B=0 2.A square=B square 3.A-B=0 4.Asquare+B square=0

**Q.**Let Tn be the number of all possible triangles formed by joining vertices of n-sided regular polygon. If Tn+1−Tn=10, then the value of n is :

- 7
- 5
- 10
- 8

**Q.**Let f(x)=tanx, g(x)=square root of (1-x^2) then g(f(x)) is

**Q.**

Simplify and express each of the following in exponential form:${25}^{4}\xf7{5}^{3}$

**Q.**

If$a,bandc$ are three unequal numbers such that $a,bandc$are in A.P and $b-a,c-banda$are in G.P., then $a:b:c$ is

$1:2:3$

$2:3:1$

$1:3:2$

$3:2:1$

**Q.**The number of ordered pairs (r, k) for which 6⋅35Cr=(k2−3)⋅36Cr+1, where k is an integer, is :

- 4
- 6
- 2
- 3

**Q.**

If $[a\times bb\times cc\times a]=\lambda {[abc]}^{2}$, then $\lambda $is equal to

$0$

$1$

$2$

$3$

**Q.**A natural number is selected at random from 1 to 1000. Then the probability that non-zero digits appear at most once is

- 11500
- 7250
- 4125
- 17750

**Q.**

If ${}^{2n}{C}_{3}:{}^{n}{C}_{2}=44:3$, then for which of the following values of $r$, the value of ${}^{n}{C}_{r}$ will be $15$?

$r=3$

$r=4$

$r=6$

$r=5$

**Q.**If 20∑i=1( 20Ci−120Ci+ 20Ci−1)3=k21, then k equals :

- 50
- 100
- 200
- 400

**Q.**

A student has to answer 10 questions, choosing at least 4 from each of part A and part B. If there are 6 questions in part A and 7 in part B, in how many ways can the student choose 10 questions ?

**Q.**

If ${}^{n+2}{C}_{8}:{}^{n-2}{P}_{4}=\frac{57}{16}$ then $n$ is equal to ?

$19$

$2$

$20$

$5$

**Q.**

Four couples (husband and wife) decided. Find the number of different committees that can be formed in which no couple finds a place.

$12$

$14$

$16$

$24$

**Q.**The number of 4 digit numbers that can be formed with 0, 1, 2, 3, 5 which are divisible by 2 or 5 is

- 84
- 60
- 53
- 120

**Q.**

If 43Cr−6=43C3r+1, then the value of r is

12

8

6

10

14

**Q.**

If the letters of the word ‘SACHIN’ are arranged in all possible ways and these words are written in dictionary order, then the word "SACHIN’ appears at serial number?

$600$

$601$

$602$

$603$

**Q.**

If ${}^{n}C_{r}$ denotes the number of combinations of $n$ things taken $r$ at a time, then the expression ${}^{n}C_{r+1}+{}^{n}C_{r-1}+2.{}^{n}C_{r}$ equals?

${}^{n+2}C_{r}$

${}^{n+2}C_{r+1}$

${}^{n+1}C_{r}$

${}^{n+1}C_{r+1}$

**Q.**Let a cricket player played n (n>1) matches during his career. If Tr represents the runs made by the player in rth match such that T1=6 and Tr=3Tr–1+6r for 2≤r≤n, then the runs scored by him in 100 matches is

- 35[4⋅6100−5⋅3100−1]
- 25[4⋅6100−5⋅3100+1]
- 35[4⋅699−5⋅399+1]
- 35[4⋅6100−5⋅3100+1]

**Q.**The value of ∑10r=1r.rPr =

- 11!
- 11! - 1
- 11! + 1
- 11! - 11

**Q.**

In an examination, a student has to answer 4 questions out of 5 questions ; questions 1 and 2 are however compulsory, Determine the number of ways in which the student can make the choice.

**Q.**

The number which should be added to the numbers $2,14,62$ so that the resulting numbers may be in G.P., is

$1$

$2$

$3$

$4$

**Q.**

In a $12$-storey house ten people enter a lift cabin. It is known that they will leave the lift in pre-decided groups of $2,3,$ and $5$ people at different storeys. The number of ways they can do so if the lift does not stop at the second storey is

$78$

$112$

$720$

$132$

**Q.**The number of ways of selecting two squares on a chess board such that they have a side in common is

- 228
- 112
- 108
- 110