AMC10 2006 A

AMC10 2006 A · Q18

AMC10 2006 A · Q18. It mainly tests Basic counting (rules of product/sum), Permutations.

A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?

某州的车牌由 4 个数字（不一定不同）和 2 个字母（也不一定不同）组成。这六个字符可以以任意顺序出现，但两个字母必须紧挨着。可能有多少种不同的车牌？

(A) $10^4 \cdot 26^2$ $10^4 \cdot 26^2$

(B) $10^3 \cdot 26^3$ $10^3 \cdot 26^3$

(D) $10^2 \cdot 26^4$ $10^2 \cdot 26^4$

(E) $5 \cdot 10^3 \cdot 26^3$ $5 \cdot 10^3 \cdot 26^3$

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) Since the two letters have to be next to each other, think of them as forming a two-letter word $w$. So each license plate consists of 4 digits and $w$. For each digit there are 10 choices. There are $26 \cdot 26$ choices for the letters of $w$, and there are 5 choices for the position of $w$. So the total number of distinct license plates is $5 \cdot 10^4 \cdot 26^2$.

（C）由于这两个字母必须相邻，可以把它们看作组成一个由两个字母构成的“单词”$w$。因此，每个车牌由 4 个数字和 $w$ 组成。每个数字有 10 种选择。$w$ 的字母共有 $26 \cdot 26$ 种选择，而 $w$ 的位置有 5 种选择。所以不同车牌的总数是 $5 \cdot 10^4 \cdot 26^2$。

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