Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Permutations and Combinations

Short Answer Type

41. Find L.C.M. and H.C.F. of 2 x 2!, 4! and 3 x 5!

131 Views

42.
Show that :

$space space space space space space space fraction numerator left parenthesis 2 straight n right parenthesis factorial over denominator straight n factorial end fraction space equals space left square bracket 1 times 3 times 5 times........... space left parenthesis 2 straight n minus 1 right parenthesis right square bracket 2 to the power of straight n$

L.H.S. = $fraction numerator left parenthesis 2 straight n right parenthesis factorial over denominator straight n factorial end fraction equals fraction numerator 1 times 2 times 3 times 4 times 5 times..... left parenthesis 2 straight n minus 1 right parenthesis space left parenthesis 2 straight n right parenthesis over denominator straight n factorial end fraction$

= $fraction numerator open square brackets 1 times 3 times 5 times....... left parenthesis 2 straight n minus 1 right parenthesis close square brackets space left square bracket 2 times 4 times space.......... left parenthesis 2 straight n right parenthesis over denominator straight n factorial end fraction$

= $fraction numerator left square bracket 1 times 3 times 5 times space..... left parenthesis 2 straight n minus 1 right parenthesis right square bracket space cross times space 2 to the power of straight n cross times left square bracket 1 times 2 times......... straight n right square bracket over denominator straight n factorial end fraction$

= $space space space fraction numerator left square bracket 1 times 3 times 5 times........ left parenthesis 2 straight n minus 1 right parenthesis space cross times space 2 to the power of straight n space cross times space straight n factorial over denominator straight n factorial end fraction space equals space left square bracket 1 times 3 times 5 times....... space left parenthesis 2 straight n minus 1 right parenthesis right square bracket space 2 to the power of straight n$

= R.H.S.

∴ L.H.S. = R.H.S.

Hence, $space space fraction numerator left parenthesis 2 straight n right parenthesis factorial over denominator straight n factorial end fraction space equals space left square bracket 1 times 3 times 5 times....... left parenthesis 2 straight n minus 1 right parenthesis right square bracket 2 to the power of straight n$