Combinatorics | Number Theory Answers

How many numbers between 8 and 840 have a remainder of 3 when divided by 7

A cricket team of 11 players is to be selected from two groups of 6 and 8 players. In how many ways can the selection be made so that at least 4 players are taken from the group of 6.

A theater has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and each succesive row of seats has two more seats in it than previous row

a) Calculate the number of seats in the 20th row.

b) calculate the total number of seats

Let U={-8,-7,....1,0,1,.....12}
A={x=-2<x<3}
B={x:-8<x<5}
C={x:0<x<12}
Find:
1.AUB
2.ANB
3.A
4.BNA

Use the fact that nCk equals n!/k!(n-k)! to express in factorials
1) The coefficient "u" of x^n in the expansion of (1+x)^2n
2) the coefficient "v" of x^n in the expansion of (1+x)^2n-1, Hence show that u=2v

Find the coefficient of x^6 in the expansion of (1-3x)(1+2x)^9 using the general term: t= nCr a^n-r b^r

The number of letters in the language of a weird country is 5 and no one in that
country uses more than 3 letters to make a word. What is the highest number of
words one can make in that language?

Prove that $\sum_{i=1}^s a_i=2^{n}-1-\sum_{i=s+1}^n a_i=2^s-1. Note that $\sum_{i=1}^n=2^n-1.

Give an example of a LPP with more than one optimal solution

A company produces their products P,Q and R from raw materials A,B and C .
To produce one unit of the product P, 2 units of A, 5 units of B and 4 units of C are
required. To produce one unit of the product Q, 1 unit of A, 1 unit of B and 2 units
of C are required. To produce one unit of the product R, 1 unit of A, 1 unit of B and ,
1 unit of C are required. Profits per unit of the products P,Q and R are Rs.10, Rs 5
and Rs. 4 respectively. The company has 10 units of A, 20 units of B and 20 units of
C. Formulate the problem of maximization of profit as a LPP.