First Principles AI

Questions

Question 1
Evaluate the following:
(i)
${}^{50}C_{30}$
(ii)
${}^{1000}C_0$
(iii)
${}^{10}C_7$
(iv)
${}^{20}C_{17}$
Solution Theory
Question 2
(i)
If ${}^3C_2 : {}^nC_2 = 15 : 1$ , find $n$ .
(ii)
If ${}^nP_r = 120$ and ${}^nC_r = 20$ , find $r$ .
Solution Theory
Question 3
Find the values of $n$ and $r$ , when
(i)
${}^nC_r = 56$ , ${}^nP_r = 336$
(ii)
${}^{n-1}C_{r-1} : {}^nC_r : {}^nC_r = 1 : 3 : 7$
Solution Theory
Question 4
Prove that
(i)
${}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r$
(ii)
$r \cdot {}^nC_r = (n-r+1) \cdot {}^nC_{r-1}$
Solution Theory
Question 5
Prove that product of $r$ consecutive integers is divisible by $r!$ .
Solution Theory
Question 6
In how many ways can five subjects be selected out of eight subjects to select a course programme?
Solution Theory
Question 7
Find the number of possible arrangements of vowel letters from the English alphabet?
Solution Theory
Question 8
In how many ways 3 dishes of Desi foods and 2 dishes of Chinese foods be selected from 6 dishes of desi foods and 8 dishes of Chinese foods?
Solution Theory
Question 9
From a standard deck of 52 playing cards, there are 26 black cards and 26 red cards. How many different ways can eight cards be selected if 3 are black and the remaining 5 are red?
Solution Theory
Question 10
A bag contains 8 red balls and 7 green balls. Find the total number of possible ways in which five balls are selected in a way:
(i)
3 red and 2 green
(ii)
1 red and 4 green
(iii)
4 red and 1 green
(iv)
all the red balls
Solution Theory
Question 11
How many diagonals and triangles can be formed by joining the vertices of the polygon having 15 sides.
Solution Theory
Question 12
Find the number of sides of a polygon if the number of its diagonals is 104.
Solution Theory
Question 13
How many triangles can be formed by joining 15 points, 6 of which lie on the same straight line?
Solution Theory
Question 14
The members of a club are 10 boys and 8 girls. In how many ways can a committee of 6 boys and 3 girls be formed?
Solution Theory
Question 15
How many committees of 7 members can be chosen from a group of 10 persons when each committee must include 2 particular persons?
Solution Theory
Question 16
In how many ways can a cricket team of 11 players be selected out of 17 players? How many of them will include a particular player?
Solution Theory
Question 17
There are 6 men and 8 women members of a club. How many committees of seven can be formed:
(i)
with 3 women
(ii)
with at most 3 women
(iii)
with at least 5 women
Solution Theory
Question 18
There are three sections in a question paper; each section has 3 questions. A student has to solve all 5 questions, choosing at least one question from each section. In how many ways can the student make his choice?
Solution Theory
Question 19
Consider a cryptographic system that generates an 8-character password. Each character in the password can be either a lowercase letter ( $a-f$ ) or a digit ( $0-5$ ). How many passwords can be generated if each password must contain exactly 5 lowercase letters and 3 digits:
(a)
with repetition allowed
(b)
without repetition
Solution Theory
Question 20
On Defense Day, Teacher I compiles a list of 10 distinct national songs, while Teacher II prepares a separate list of 10 different national songs (with no overlap between the two lists). The principal needs to select 3 songs from Teacher I's list, and 3 songs from Teacher II's list. Determine the number of possible selection methods when:
(i)
the order/sequence of the selected songs is important.
(ii)
the order/sequence of the selected songs is not important.
Solution Theory