1
IIT-JEE 1983
MCQ (Single Correct Answer)
+1
-0.25
The coefficient of $${x^4}$$ in $${\left( {{x \over 2} - {3 \over {{x^2}}}} \right)^{10}}$$ is
A
$${{{405} \over {256}}}$$
B
$${{{504} \over {259}}}$$
C
$${{{450} \over {263}}}$$
D
none of these
2
IIT-JEE 1983
Subjective
+3
-0
If $${\left( {1 + x} \right)^n} = {C_0} + {C_1}x + {C_2}{x^2} + ..... + {C_n}{x^n}$$ then show that the sum of the products of the $${C_i}s$$ taken two at a time, represented $$\sum\limits_{0 \le i < j \le n} {\sum {{C_i}{C_j}} } $$ is equal to $${2^{2n - 1}} - {{\left( {2n} \right)!} \over {2{{\left( {n!} \right)}^2}}}$$
3
IIT-JEE 1983
Subjective
+2
-0
Use mathematical Induction to prove : If $$n$$ is any odd positive integer, then $$n\left( {{n^2} - 1} \right)$$ is divisible by 24.
4
IIT-JEE 1983
Subjective
+2
-0
m men and n women are to be seated in a row so that no two women sit together. If $$m > n$$, then show that the number of ways in which they can be seated is $$\,{{m!(m + 1)!} \over {(m - n + 1)!}}$$
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