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Binomial Theorem

Short Answer Type

71.

Find the 4^th term from the end in the expansion of $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

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72.

Find the 11^th term from the end in the expansion of $open parentheses 2 straight x minus 1 over straight x squared close parentheses to the power of 25$

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73. Prove that in the expansion of (a + b)ⁿ, the co-efficients of terms equidistant from the beginning and end are the same.

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Long Answer Type

74.

Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $open parentheses fourth root of 2 plus fraction numerator 1 over denominator fourth root of 3 end fraction close parentheses to the power of straight n$ is $square root of 6 colon 1.$

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Short Answer Type

75. Find the co-efficient of

straight x to the power of 5 space in space left parenthesis straight x plus 3 right parenthesis to the power of 8

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76. Find the co-efficient of:

straight a to the power of 5 straight b to the power of 7 space in space left parenthesis straight a minus 2 straight b right parenthesis to the power of 12

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77. Find the positive value of m for which the co-efficient of x² in the expansion of (1 + x)^m is 6.

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78. In the expansion of (1 + x)ⁿ, the co-efficients of x^p-1 and of x^q-1 are equal. Show that p + q = n + 2, p ≠ q.

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79. In the expansion of (1 + a)^m+n , prove that co-efficient of a^m and aⁿ are equal.

Co-efficient of $straight a to the power of straight m$ in the expansion of $space space space space space left parenthesis 1 plus straight a right parenthesis to the power of straight m plus straight n end exponent space equals space straight C presuperscript straight m plus straight n end presuperscript subscript straight n$ ...(i)

Also, co-efficient of $space straight a to the power of straight n$ in the expansion of $space space space space space space left parenthesis 1 plus straight a right parenthesis to the power of straight m plus straight n end exponent space equals space straight C presuperscript straight m plus straight n end presuperscript subscript straight n$

$equals space straight C presuperscript straight m plus straight n end presuperscript subscript left parenthesis straight m plus straight n right parenthesis minus straight n end subscript$ (∵ $space space space space space straight C presuperscript straight n subscript straight r space equals space straight C presuperscript straight n subscript straight n minus straight r end subscript$ )

$equals space straight C presuperscript straight m plus straight n end presuperscript subscript straight m$ ...(ii)
Hence, from (i) and (ii), the co-efficient of $straight a to the power of straight m$ is equal to the co-efficient of $straight a to the power of straight n.$