問82
頻出度★★☆
次の体系をもつ電話番号において，80億個の番号を創出したい。番号の最低限必要な桁数は幾つか。ここで，桁数には“020”を含むこととする。
0 2 0 □ □ … □
- 各桁に0～9のいずれかを指定（“020”に続く先頭の□を除く，それ以降の各□） - 1～3及び5～9のいずれかを指定（“020”に続く先頭の□）
令和元年度 秋期 ITパスポート試験 問82
テクノロジ系基礎理論 › 応用数学
This question's explanation and your answers so far.
解説
問 82
Incorrectア
With 11 digits, the variable digits number 11−3＝8, so the total is only 8×10^7＝80 million, far short of 8 billion.
Incorrectイ
With 12 digits, the variable digits number 12−3＝9, so the total is 8×10^8＝800 million, only one tenth of 8 billion and thus insufficient. This value overlooks that adding one more variable digit multiplies the total by 10.
- ウ
正解 A telephone number consists of the fixed leading “020” (3 digits) followed by variable □ digits. Among the variable digits, the first □ has 8 choices (1 to 3 and 5 to 9), and each □ after it has 10 choices (0 to 9). Let n be the number of variable digits; then the total count of numbers that can be made is 8×10^(n−1). This only needs to be at least 8 billion = 8×10^9, so from 8×10^(n−1) ≧ 8×10^9 we get 10^(n−1) ≧ 10^9, i.e. n−1 ≧ 9, so n ≧ 10. The minimum number of variable digits is 10 (at which point the total is exactly 8×10^9 = 8 billion). Since the digit count includes the fixed 3 digits of “020”, the minimum number of digits required is 3＋10 = 13. Therefore the correct answer is ウ.
Incorrectエ
With 14 digits, the variable digits number 14−3＝11, so the total is 8×10^10＝80 billion, far exceeding 8 billion. It meets the requirement but is not the “minimum number of digits required,” since 13 digits already suffice, making 14 excessive.
ここがポイント
- The product rule: multiply the number of choices at each digit to get the total. The first variable □ has 8 choices and each remaining digit has 10, so the total is 8×10^(number of variable digits −1).
- The “minimum number of digits required” means choosing the smallest digit count whose total reaches or exceeds the required value (8 billion), while remembering that the fixed 3 digits of “020” must be included in the count.
用語