Assertion :A fair coin is tossed fixed number of times. If probability of getting 7 heads is equal to probability of getting 9 heads then probability of getting 2 heads is equal to $\frac{15}{2^{12}}$ . Reason: If $^{n} C_{x} =^{n} C_{y}$ then either $x = y$ or $x + y = n$ and $P (X = r) =^{n} C_{r} p^{r} q^{n - r}$ .

Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

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Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion

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Assertion is true but Reason is false

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Assertion is false but Reason is true

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Solution

The correct option is D Assertion is false but Reason is true
Let $n$ be the total number of tosses. & $X$ be the number of times head occurs.
$∴ p = \frac{1}{2} = q$
Now given $P (7) = P (9)$
$\Rightarrow^{n} C_{7} {(\frac{1}{2})}^{7} {(\frac{1}{2})}^{n - 7} =^{n} C_{9} {(\frac{1}{2})}^{9} {(\frac{1}{2})}^{n - 9}$
$\Rightarrow^{n} C_{7} =^{n} C_{9}$
$\Rightarrow n = 16$
$∴ P (2) =^{16} C_{2} {(\frac{1}{2})}^{2} {(\frac{1}{2})}^{14} = \frac{15}{2^{13}}$
$∴$ Assertion (A) is false but Reason (R) is true.

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When a polynomial p(x) is divided by (x − α), then the remainder is p(α).

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Assertion : Two dice are thrown simultaneously. There are 11 possible outcomes (2,3,4,5,6,7,8,9,10,11,12) and each of them has a probability $\frac{1}{11}$ .
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Assertion :A fair coin is tossed fixed number of times. If probability of getting 7 heads is equal to probability of getting 9 heads then probability of getting 2 heads is equal to 15212 . Reason: If nCx=nCy then either x=y or x+y=n and P(X=r)=nCrprqn−r.

The correct option is D Assertion is false but Reason is trueLet n be the total number of tosses. & X be the number of times head occurs.∴p=12=qNow given P(7)=P(9)⇒nC7(12)7(12)n−7=nC9(12)9(12)n−9⇒nC7=nC9⇒n=16∴P(2)=16C2(12)2(12)14=15213∴ Assertion (A) is false but Reason (R) is true.