• Number of Basic Variables in Transportation Problem:
• In a standard transportation problem with $X$ sources (origins) and $Y$ destinations:
• The problem has $X + Y$ linear constraint equations (row sums and column sums).
• Because the total supply equals total demand ($\sum a_i = \sum b_j$), one constraint equation is linearly dependent (redundant).
• Therefore, the rank of the constraint coefficient matrix is $X + Y - 1$.
• Hence, any non-degenerate basic feasible solution contains exactly $X + Y - 1$ basic variables (allocated cells).
• Conclusion: Total basic variables is X + Y - 1 (Option 1).

Correct option: (1) X+Y¥-1

• Determining the Optimal Number of Clusters:
• In cluster analysis and principal component analysis, the Scree plot (introduced by Cattell) graphs the eigenvalues or within-cluster sum of squares (WSS) against the number of components/clusters.
• The "elbow" point where the curve flattens out (the transition from steep cliff to gentle scree) identifies the optimal number of clusters/components.
• Other Plots:
• Scatter plot: Displays bivariate relationships.
• Biplot: Superimposes observations and variables in PCA.
• Normal probability plot: Tests distributional normality.
• Conclusion: Judged based on the Scree plot (Option 1).

Correct option: (1) Sctee plot