Track / Stage 3 / 94 of 260 #94 of 2000

Problem 94

AMC 10/12, early questions

Combinatorics Difficulty 3.3 Find the answer

The Shan City Agricultural Science Research Institute planted 5 different types of seeds in 5 adjacent experimental fields. Among them, the seeds of types A and B are required to be planted in two adjacent fields, and neither can be planted in the fields at the ends. The number of different planting methods is ( ).

A: $12$

B: $24$

C: $36$

D: $48$

Your answer

Official solution

First, select 2 out of the 3 different types of seeds, excluding types A and B, to plant at the two ends. Then, bundle types A and B together with the remaining type for a full permutation planting in the three fields not including the ends. Therefore, there are $A_{3}^{2}A_{2}^{2}A_{2}^{2}=24$ methods,

Thus, the answer is: $\boxed{B}$ .

First, select 2 out of the 3 different types of seeds, excluding types A and B, to plant at the two ends. Then, bundle types A and B together with the remaining type for a full permutation planting in the three fields not including the ends. According to the principle of step-by-step counting, this can be determined.

This question examines the application of counting principles, the ability to perform operations and solve problems, and the idea of discussing different cases. Pay attention to accumulating problem-solving methods. It is considered a medium-difficulty question.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.