<s>[INST] Finding the remainder when $1.1!+2.2!+3.3!+ ... +10.10! +2$ is divided by $11!$ Find the remainder when $1.1!+2.2!+3.3!+ ... +10.10! +2$ is divided by $11!$
An attempt: Rearranging:
$$\frac{1}{11!}+\frac{2.2!}{11}+\frac{3.3!}{11} \cdots +\frac{10.10!}{11}+\frac{2}{11!}$$
$$\frac{1}{11!}+\frac{2}{11!}+\frac{2.2!}{11}+\frac{3.3!}{11} \cdots +\frac{10.10!}{11}$$
$$\frac{1}{11!}+\frac{(2+1)2!}{11!}+\frac{3.3!}{11} \cdots +\frac{10.10!}{11}$$
$$\frac{1}{11!}+\frac{3.2!}{11!}+\frac{3.3!}{11} \cdots +\frac{10.10!}{11}$$
$$\frac{1}{11!}+\frac{(3+1)3!}{11!}+\frac{3.3!}{11} \cdots +\frac{10.10!}{11}$$
$$.$$
$$.$$
$$.$$
$$\frac{1}{11!}+\frac{(10+1)10!}{11!}$$
$$\frac{11!+1}{11!}$$
What is the remainder?
 [/INST] $$\sum_{k=0}^n k.k!=(n+1)!$$.this summation should be used in your question
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