Question

Assertion :Number of ways of selecting 10 objects from 42 objects of which 21 objects are identical and remaining objects are distinct is $2^{20}$ Reason: ${}^{42}{C}_0{+}^{42}{C}_1{+}^{42}{C}_2+...{+}^{42}{C}_{21}=2^{41}$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Number of objects from 21 different objects	Number of objects from 21 identical objects	Number of ways of selections
10	0	${}^{21}{C}_{10}\times 1$
9	1	${}^{21}{C}_9\times 1$
:	:	:
0	10	${}^{21}{C}_0\times 1$

Thus, total number of ways of selection is
${}^{21}{C}_0{+}^{21}{C}_1{+}^{21}{C}_2+....{+}^{21}{C}_{10}=2^{20}$
Statement 2 is false, as given series is not exact half series.