Mathematics - Probability Question with Solution | TestHub

MathematicsProbabilityGeneral (Event SS, Definitions, Odd against/Favour)Medium2 minQB

MathematicsMediuminteger

If k be the number of 3 digit natural numbers, having sum of their digits atleast 10 , then the value of $\frac{k - 35}{100}$ is

Answer:

Solution:

We have to calculate number of solution of $a + b - c > 9, 1 \leq a \leq 9, 0 \leq b, c \leq 9$

$a + b + c + d = 9, d \geq 0$

Number of solutions is co-efficient of $t^{9}$ in

$(t + t^{2} + \dots . + t^{9}) (1 + t + \dots .. + t^{7})^{2} (1 + t + t^{2} + \dots .. t^{9})$

$= coe ff i c i e n t o f t^{8} in (1 - t^{9}) (1 - t^{10})^{2} (1 - t)^{- 4}$

$=$ coefficient of $t^{8}$ in $(1 - t)^{- 4} =^{8 + 4 - 1} C_{4 - 1} =^{11} C_{3} = 165$

So, required number of natural numbers $900 - 165 = 735$

$∴ \frac{k - 35}{100} = 7$

Stream:JEESubject:MathematicsTopic:ProbabilitySubtopic:General (Event SS, Definitions, Odd against/Favour)

⏱ 2mℹ️ Source: QB

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