Question

Draw the linear arrangement: "Six swimmers--H, J, K, L, N, and P--are assigned to six swimming lanes numbered 1 through 6. Exactly one swimmer is assigned to each lane. The lane assignments conform to the following conditions:

Swimmer K is assigned a lower-numbered lane than is swimmer J.
Swimmer P is assigned a lower-numbered lane than is swimmer K

Answer 1

K is in a lower-numbered lane than J, and P is in a lower-numbered lane than K. The linear arrangement is K < J and P < K.

Step-by-step explanation:

The given conditions indicate that Swimmer K has a lower-numbered lane than Swimmer J. Representing this relationship as K < J, we establish the relative order of lanes for K and J. Additionally, the conditions state that Swimmer P has a lower-numbered lane than Swimmer K, leading to P < K. Combining these relationships, we obtain the linear arrangement: K < J and P < K.

To illustrate this linear arrangement, we represent it graphically:

\[ \underline{1} \ \underline{2} \ \underline{3} \ \underline{4} \ \underline{5} \ \underline{6} \]

\[ \underline{K} \ \underline{J} \ \underline{P} \]

Here, the numbers below the lanes indicate their order, and the letters above represent the corresponding swimmers. K is in a lower-numbered lane than J, and P is in a lower-numbered lane than K, satisfying the given conditions. This linear arrangement visually depicts the assigned lanes for swimmers K, J, and P.

In conclusion, the linear arrangement is K < J and P < K, meeting the specified conditions for the assignment of lanes to swimmers.