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Yogita Ingle 5 years ago
The number of proper subsets of any set is given by the formula 2n -1 where n is the number of elements in the set.
Here n = 5
25 -1 = 32 - 1 = 31
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Yogita Ingle 5 years ago
A permutation is defined as an arrangement in a definite order of a number of objects taken some or all at a time. Counting permutations are merely counting the number of ways in which some or all objects at a time are rearranged. The convenient expression to denote permutation is defined as “ nPr ”.
The permutation formula is given by,
| Pr = n!/(n-r)! ; 0 ≤ r ≤ n |
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Gaurav Seth 5 years ago
We know that, Three lines are said to be concurrent, if they pass through a common point which also means that the point of intersection of any two lines always lies on the third line.
Now, the three given equations of line are:
2x + y − 3 = 0 ... (1)
5x + ky − 3 = 0 ... (2)
and 3x − y − 2 = 0 ... (3)
Firstly, we will solve the equations (1) and (3) and find out the value of x and y. The corresponding value of x and y will also satisfy the equation (2) as all the given lines are concurrent.
Therefore, on adding (1) and (3), we get
2x + y − 3 + 3x − y − 2 = 0
⇒ 5x - 5 = 0
⇒ x = 1
On putting value of x in (1), we get
2(1) + y − 3 =0
⇒ y - 1 = 0
⇒ y = 1
So, the point of intersection of two lines is (1, 1).
Now, this point will also satisfy the equation (2).
Therefore, on putting (x, y) = (1, 1) in (2), we get
5(1) + k(1) − 3 = 0
⇒ k + 2 = 0
⇒ k = - 2
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For n = 1 ,
p(k) is true
Let for n = k ,p(k) is true
For n = k+1:
(k+1)(k+2)(k+3)
k(k+1)(k+2)(k+3)/k divisible by 6
So by mathematical induction , statement is true for all n€N
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Sajid Md 5 years ago
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