Problem statement:

問題陳述：

Given a length n, count the number of strings of length n that can be made using 'a', 'b' and 'c' with at-most one 'b' and two 'c's allowed.

給定長度n ，計算可以使用'a' ， 'b'和'c'且長度最多為'b'和兩個'c'的長度為n的字符串的數量。

Example:

例：

Input: n=1Output:3Possible strings are:"a", "b", "c"Input: n=2Output:8Possible strings are:"aa", "ab", "ac", "ba", "ca", "bc", "cb", "cc"

Solution:

解：

String alphabets are only {a, b, c}

字符串字母僅為{a，b，c}

Length of string is n. (n>0)

字符串的長度是n。 (n> 0)

Let's consider what can be the possible cases

讓我們考慮一下可能的情況

String is only built with 'a', i.e., n 'a' forms the string.

字符串僅使用'a'構建，即n'a '構成字符串。

Count of such string is: 1

該字符串的計數為：1

String built with one 'b' & n-1 'a'

串建有一個“B”及n-1個 “A”

Count of such string is:

該字符串的計數為：

(n/1)=n

(n / 1)= n

One

之一

'b' can be placed at any of n positions, that's why n number of such strings

'b'可以放置在n個位置中的任何位置，這就是為什么n個這樣的字符串

String built with one 'b', one 'c' and (n-2) 'a'

用一個'b' ，一個'c'和(n-2) 'a'構建的字符串

Count of such string

這樣的字符串數

(n/2)*2=n*(n-1)

(n / 2)* 2 = n *(n-1)

One

之一

'b' and one 'c' can take any of two places out of n and any of 'b' & 'c' can comes first.

'b'和一個'c'可以從n中占據兩個位置中的任何一個，并且'b'和'c'中的任何一個都可以排在第一位。

String built with one 'b', two 'c' and (n-3) 'a'

用一個'b' ，兩個'c'和(n-3) 'a'構建的字符串

Count of such string

這樣的字符串數

(n/3)*3=n*(n-1)*(n-2)/2

(n / 3)* 3 = n *(n-1)*(n-2)/ 2

One

之一

'b' and two 'c' can take any of three places out of n and there are 3 combinations possible between one 'b' & two 'c'.

“b”和2“c”的可以采取任何的三個地方出n和有3種組合之一“B”及2“C”之間的可能。

String built with two 'c' and (n-2) 'a'

用兩個'c'和(n-2) 'a'構建的字符串

Count of such string

這樣的字符串數

(n/2)=n*(n-1)/2

(n / 2)= n *(n-1)/ 2

Two

二

'c' can take any two of n places.

'c'可以取代n位中的任何兩個。

String built with one 'c' and (n-1) 'a'

用一個'c'和(n-1) 'a'構建的字符串

Count of such string

這樣的字符串數

(n/1)=n

(n / 1)= n

One

之一

'c' can take any of one places out of n.

'c'可以取n中的任何一位。

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Example with explanation

帶說明的例子

Let n=2Case 1: String is only built with 'a', i.e., n 'a' forms the string"aa"Total under this category: 1Case 2: String built with one 'b' & n-1 'a'"ab""ba"Total under this category: 2//(n)Case 3: String built with one 'b', one 'c' and (n-2) 'a'"bc""cb"Total under this category: 2//(n*(n-1))Case 4: String built with one 'b', two 'c' and (n-3) 'a'No string in this categoryTotal under this category: 0Case 5: String built with two 'c' and (n-2) 'a'"cc"Total under this category: 1//(n*(n-1)/2)Case 6: String built with one 'c' and (n-1) 'a'"ac""ca"Total under this category: 2//(n)Total no of strings possible: 1+2+2+0+1+2=8

C++ implementation

C ++實現

#include <bits/stdc++.h> using namespace std;int find(int n){//total no of string possible(for details check solution part)return 1+(n)+n*(n-1)+n*(n-1)*(n-2)/2+n*(n-1)/2+n; }int main() {int n;cout<<"enter length of string\n";cin>>n;cout<<"Number of string possible under ";cout<<"constraints is : "<<find(n)<<endl;return 0; }

Output

輸出量

First run: enter length of string 2 Number of string possible under constraints is : 8Second run: enter length of string 4 Number of string possible under constraints is : 39

翻譯自: https://www.includehelp.com/icp/count-of-strings-that-can-be-formed-using-a-b-and-c-under-given-constraints.aspx

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