AMC10 2007 B

AMC10 2007 B · Q8

AMC10 2007 B · Q8. It mainly tests Basic counting (rules of product/sum), Digit properties (sum of digits, divisibility tests).

On the trip home from the meeting where this AMC10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form bbcac, where $0 \leq a < b < c \leq 9$, and $b$ was the average of $a$ and $c$. How many different five-digit numbers satisfy all these properties?

在构造本AMC10的会议回家的路上，Contest Chair注意到他的机场停车收据上的数字形式为bbcac，其中$0 \leq a < b < c \leq 9$，且$b$是$a$和$c$的平均数。满足所有这些性质的不同五位数有多少个？

(A) 12 12

(B) 16 16

(D) 20 20

(E) 24 24

Answer

Correct choice: (D)

正确答案：（D）

Solution

Once a and c are chosen, the integer b is determined. For a = 0, we could have c = 2, 4, 6, or 8. For a = 2, we could have c = 4, 6, or 8. For a = 4, we could have c = 6 or 8, and for a = 6 the only possibility is c = 8. Thus there are 1 + 2 + 3 + 4 = 10 possibilities when a is even. Similarly, there are 10 possibilities when a is odd, so the number of possibilities is 20.

一旦选择a和c，整数b就确定了。对于a=0，可有c=2,4,6或8。对于a=2，可有c=4,6或8。对于a=4，可有c=6或8，对于a=6只有c=8。这样，当a为偶数时有1+2+3+4=10种可能。类似地，当a为奇数时也有10种可能，因此总数为20。

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