D Playing cards 2 n (n - 1) (n - 2) (3)(2)(1) can be simplified as A n raised to n B P(n, r) C n! D C(n, r) 15
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The expression "2n (n - 1) (n - 2) (3J(2)(1)" can be simplified as follows: - First, simplify the expression inside the parentheses: (3J(2)(1)) = 3! = 3 x 2 x 1 = 6. - Now, the expression becomes: 2n (n - 1) (n - 2) (6). Show more…
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Suppose for cach $n \in \mathbf{N}$. $\left(1^{2}-a_{1}\right)+\left(2^{2}-a_{2}\right)+\ldots+\left(n^{2}-a_{n}\right)=\frac{1}{3} n\left(n^{2}-1\right)$ then $a_{n}$ cquals (a) $n$ (b) $n-1$ (c) $n+1$ (d) $2 n$
$a_{n}=n !$ (Hint: Recall that $n !=n(n-1)(n-2) \cdots 2 \cdot 1$ )
Sequences and Infinite Series
An Overview
The number of ways in which we can arrange the $2 n$ Ietters $x_{1}, x_{2}, \cdots, x_{m} y_{1}, y_{2}, \cdots, y_{n}$ in a line so that the suffixes of letters $x$ and those of $y$ are respectively in ascending order of magnitude is (a) ${ }^{2 n} C_{m}$ (b) $\left({ }^{\prime \prime} C_{0}\right)^{2}+\left({ }^{\circ} C_{1}\right)^{2}+\cdots+\left({ }^{e} C_{n}\right)^{2}$ (c) $\frac{1}{2}(2 n) !$ (d) $\frac{1}{4}(2 n) !$
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