Question
If 9P5 + 5 9P4 = 10Pr , then r =
- 4
- 5
- 9
- 10
Hint:
Use the formula ![P presuperscript n subscript r space equals space fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial end fraction](data:image/png;base64,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)
The correct answer is: 5
Given : ![P presuperscript 9 subscript 5 space plus space 5 space cross times P presuperscript 9 subscript 4 space equals space P presuperscript 10 subscript r](data:image/png;base64,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)
Using Formula :
![P presuperscript n subscript r space equals space fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial end fraction](data:image/png;base64,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)
![fraction numerator 9 factorial over denominator left parenthesis 9 minus 5 right parenthesis factorial end fraction space plus space 5 cross times fraction numerator 9 factorial over denominator left parenthesis 9 minus 4 right parenthesis factorial end fraction space equals space fraction numerator 10 factorial over denominator left parenthesis 10 minus r right parenthesis factorial end fraction](data:image/png;base64,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)
![fraction numerator 9 factorial over denominator 4 factorial end fraction space plus space 5 space cross times space fraction numerator 9 factorial over denominator 5 factorial end fraction space equals space fraction numerator 10 cross times 9 factorial over denominator left parenthesis 10 minus r right parenthesis factorial end fraction](data:image/png;base64,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)
Dividing both sides by 9!
![fraction numerator 1 over denominator 4 factorial end fraction plus 5 cross times fraction numerator 1 over denominator 5 factorial end fraction space equals space fraction numerator 10 over denominator left parenthesis 10 minus r right parenthesis factorial end fraction](data:image/png;base64,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)
![fraction numerator 5 over denominator 5 factorial end fraction plus space fraction numerator 5 over denominator 5 factorial end fraction equals space fraction numerator 10 over denominator left parenthesis 10 minus r right parenthesis factorial end fraction
fraction numerator 10 over denominator 5 factorial end fraction space equals space fraction numerator 10 over denominator left parenthesis 10 minus r right parenthesis factorial end fraction
E q u a t i n g space t h e space d e n o m i n a t o r s
5 factorial space equals space left parenthesis 10 minus r right parenthesis factorial
5 space equals space 10 space minus space r
r space equals space 5](data:image/png;base64,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Related Questions to study
The number of proper divisors of
.
. 15r is-
The number of proper divisors of
.
. 15r is-
If
have a common factor then 'a' is equal to
If
have a common factor then 'a' is equal to
A block C of mass is moving with velocity and collides elastically with block of mass and connected to another block of mass through spring constant .What is if is compression of spring when velocity of is same ?
A block C of mass is moving with velocity and collides elastically with block of mass and connected to another block of mass through spring constant .What is if is compression of spring when velocity of is same ?
If
then ascending order of A, B, C.
If
then ascending order of A, B, C.
The number of different seven digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is-
We know that there is not much difference between permutation and combination. Permutation is the way or method of arranging numbers from a given set of numbers such that the order of arrangement matters. Whereas combination is the way of selecting items from a given set of items where order of selection doesn’t matter. Both the word combination and permutation is the way of arrangement. Here, we will not use permutation because the order of toys is not necessary.
The number of different seven digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is-
We know that there is not much difference between permutation and combination. Permutation is the way or method of arranging numbers from a given set of numbers such that the order of arrangement matters. Whereas combination is the way of selecting items from a given set of items where order of selection doesn’t matter. Both the word combination and permutation is the way of arrangement. Here, we will not use permutation because the order of toys is not necessary.
The centre and radius of the circle
are respectively
The centre and radius of the circle
are respectively
The centre of the circle
is
The centre of the circle
is
The equation of the circle with centre at
, which passes through the point
is
The equation of the circle with centre at
, which passes through the point
is
The foot of the perpendicular from
on the line
is
The foot of the perpendicular from
on the line
is
The foot of the perpendicular from the pole on the line
is
The foot of the perpendicular from the pole on the line
is
The equation of the line parallel to
and passing through
is
The equation of the line parallel to
and passing through
is
The line passing through the points
, (3,0) is
So here we used the concept of the equation of the line passing through two points. Here we also used the trigonometric terms to find the answer using the formulas. So the final solution is .
The line passing through the points
, (3,0) is
So here we used the concept of the equation of the line passing through two points. Here we also used the trigonometric terms to find the answer using the formulas. So the final solution is .
Statement-I : If
then A=![7 over 4](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJ9JREFUeNpjYMAO/hPAFINQIJ5NqSHiQHwYiDkoNWgTEBtQakgGENdSaog8EJ8EYhZKDToIxI7UiKVtlBrCBMQ3qBHAAVBvUQyWA3EiNQx6CcSCDKNgcIP/ZOJRMFiBDzViiAeIr1HDoJlAnEypQdZAvBcpwZIF2ID4ErQWocigDiDOQctCJAM9ID6KJS+SDEAVogo1DKJpjqdakUEfgwCGRTp0g/2KWQAAAGJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWZyYWM+PG1uPjc8L21uPjxtbj40PC9tbj48L21mcmFjPjwvbWF0aD7kU9SNAAAAAElFTkSuQmCC)
Statement-II : If
then ![p equals 2 comma q equals 3](data:image/png;base64,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)
Which of the above statements is true
Statement-I : If
then A=![7 over 4](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJ9JREFUeNpjYMAO/hPAFINQIJ5NqSHiQHwYiDkoNWgTEBtQakgGENdSaog8EJ8EYhZKDToIxI7UiKVtlBrCBMQ3qBHAAVBvUQyWA3EiNQx6CcSCDKNgcIP/ZOJRMFiBDzViiAeIr1HDoJlAnEypQdZAvBcpwZIF2ID4ErQWocigDiDOQctCJAM9ID6KJS+SDEAVogo1DKJpjqdakUEfgwCGRTp0g/2KWQAAAGJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWZyYWM+PG1uPjc8L21uPjxtbj40PC9tbj48L21mcmFjPjwvbWF0aD7kU9SNAAAAAElFTkSuQmCC)
Statement-II : If
then ![p equals 2 comma q equals 3](data:image/png;base64,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)
Which of the above statements is true
If b > a , then the equation, (x - a) (x - b) - 1 = 0, has:
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, one of the roots will be in the interval of (−α,a) and the other root will be in the interval (b,α).
If b > a , then the equation, (x - a) (x - b) - 1 = 0, has:
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, one of the roots will be in the interval of (−α,a) and the other root will be in the interval (b,α).
If
be that roots
where
, such that
and
then the number of integral solutions of λ is
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, the number of integral solutions of λ is in between
If
be that roots
where
, such that
and
then the number of integral solutions of λ is
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, the number of integral solutions of λ is in between