1. Tardigrade
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  4. Consider a sequence of 1001 terms as ( 1001 C0/1 ⋅ 2 ⋅ 3 ⋅ 4), ( 1001 C1/2 ⋅ 3 ⋅ 4 ⋅ 5), ( 1001 C2/3 ⋅ 4 ⋅ 5 ⋅ 6), ldots ( 1001 C1000/1001 ⋅ 1002 ⋅ 1003 ⋅ 1004)

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Q. Consider a sequence of $1001$ terms as
$\frac{{ }^{1001} C_{0}}{1 \cdot 2 \cdot 3 \cdot 4}, \frac{{ }^{1001} C_{1}}{2 \cdot 3 \cdot 4 \cdot 5}, \frac{{ }^{1001} C_{2}}{3 \cdot 4 \cdot 5 \cdot 6}, \ldots \frac{{ }^{1001} C_{1000}}{1001 \cdot 1002 \cdot 1003 \cdot 1004}$

Binomial Theorem

Solution:

$T_{r+1}=\frac{{ }^{1001} C_{r}}{(r+1)(r+2)(r+3)(r+4)}$
$=\frac{{ }^{1004} C_{r+4}}{1001 \cdot 1002 \cdot 1003 \cdot 1004}$ is greatest when $r=498$
$\therefore 499$ th term is greatest.