<p>There are six periods in each working day of a school. Find the number of ways in which 5 subjects can be arranged if each subject is allotted at least one period and no period remains vacant.</p>
Step-by-Step Solution
Key Concept: We must first distribute 6 periods among 5 subjects such that each subject gets at least one period (using stars and bars with constraint), then arrange the subjects in those periods. The distribution is unique: one subject gets 2 periods and four get 1 period each.
<p><strong>Step 1:</strong> Identify the constraint. We have 6 periods and 5 subjects. Each subject must get at least 1 period, and no period remains vacant.</p><p><strong>Step 2:</strong> Since we have 6 periods for 5 subjects with no gaps, by pigeonhole principle, exactly one subject gets 2 periods and the remaining four get 1 period each.</p><p><strong>Step 3:</strong> Choose which subject gets 2 periods: <strong>C(5,1) = 5 ways</strong></p><p><strong>Step 4:</strong> Once subjects are assigned their period counts, arrange all 6 periods with these subjects. This is a permutation of 6 items where one subject appears twice:</p><p>Number of arrangements = 6!/(2!×1!×1!×1!×1!) = 720/2 = 360</p><p><strong>Step 5:</strong> Total arrangements = (Ways to choose which subject gets 2 periods) × (Ways to arrange in 6 periods)</p><p>= 5 × 360 = <strong>1800</strong></p>
Correct Answer: 1800