Circular Permutations
Circular Permutations - When things are arranged in places along a line with first and last place, they form a linear permutation. So far we have dealt only with linear permutations. When things are arranged in places along a closed curve or a circle, in which any place may be regarded as the first..
Circular Permutations - When things are arranged in places along a line with first and last place, they form a linear permutation. So far we have dealt only with linear permutations. When things are arranged in places along a closed curve or a circle, in which any place may be regarded as the first..To find the nth term of a GP, whose first term is a common ratio r and number of terms is n
We observe that the index of r on the right hand side is one less than the suffix of t on the left hand side in each of the equalities. Hence t n = ar n - 1 which is the general term of the given G..
We observe that the index of r on the right hand side is one less than the suffix of t on the left hand side in each of the equalities. Hence t n = ar n - 1 which is the general term of the given G..Question 7
Question: Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Find the number of words which have atleast one letter repeated. Answer: Total number of 5 letter words from given 10 letters = 10 5 (if every letter may be ..
Cramer's rule for the solution of a system of equations in 2 variables
We recall from our earlier classes that a system of linear equation with two variables is given by This system of linear equation may have either one solution or infinitely many solutions or no solutio..
We recall from our earlier classes that a system of linear equation with two variables is given by This system of linear equation may have either one solution or infinitely many solutions or no solutio..Note:
Sum to infinity exists only when r is numerically less than 1. i.e. |r|<1..
Harmonic Mean (H.M.)
If three quantities are in harmonic progression, then the middle quantity is called the harmonic mean between the other two. Example: 1/3, 1/7, 1/11 are in H.P., then 1/7 is the middle term. Hence 1/7 is the harmonic mean between 1/3 and h..
Examples:
1. Find the sum to the series 1+2x+3x 2 +.... to n terms and to infinity when x < 1..
Proof:
From the definition of inverse of a matrix, we have (AB)(AB) - 1 = I or A - 1 (AB)(AB) - 1 = A - 1 I (Pre-multiplying both sides by A - 1 ) or (A - 1 A) B (AB) - 1 = A - 1 (Since A - 1 I = A -..
Note:
1. O!= 1 2. When n is a negative or fraction, n! is not defined...
1. O!= 1 2. When n is a negative or fraction, n! is not defined...Question 4
Let us find the number of committees of these choices in which L 1 and L 2 serve on the same committee. For type I committee, 3 men can be chosen in 1 0 C 3 ways. After having done that, we have to choose 3 ladies. But it has been decided to choose L 1 and L 2 already...
Let us find the number of committees of these choices in which L 1 and L 2 serve on the same committee. For type I committee, 3 men can be chosen in 1 0 C 3 ways. After having done that, we have to choose 3 ladies. But it has been decided to choose L 1 and L 2 already...See what our Users say :
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