Six X's have to be placed in the...

MathematicsCombinations and SelectionJEE Advanced 1978Easy

Six X's have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be done.

Answer:26

Visualized Solution (English)

Analyzing the Grid Structure

Row 1 ( $R_{1}$ ): 2 squares
Row 2 ( $R_{2}$ ): 4 squares
Row 3 ( $R_{3}$ ): 2 squares
Total squares: 8

Defining Distribution Variables

Total 'X's to place: 6
Constraint: Each row must have at least one 'X'
Let $x_{i}$ be the number of 'X's in row $R_{i}$
$x_{1} + x_{2} + x_{3} = 6$
$1 \leq x_{1} \leq 2$ , $1 \leq x_{2} \leq 4$ , $1 \leq x_{3} \leq 2$

Case 1: Distribution .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (1, 4, 1)

Case 1: $x_{1} = 1, x_{2} = 4, x_{3} = 1$
Ways: $^{2} C_{1} \times^{4} C_{4} \times^{2} C_{1} = 2 \times 1 \times 2 = 4$

Case 2: Distribution .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (1, 3, 2)

Case 2: $x_{1} = 1, x_{2} = 3, x_{3} = 2$
Ways: $^{2} C_{1} \times^{4} C_{3} \times^{2} C_{2} = 2 \times 4 \times 1 = 8$

Case 3: Distribution .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (2, 3, 1)

Case 3: $x_{1} = 2, x_{2} = 3, x_{3} = 1$
Ways: $^{2} C_{2} \times^{4} C_{3} \times^{2} C_{1} = 1 \times 4 \times 2 = 8$

Case 4: Distribution .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (2, 2, 2)

Case 4: $x_{1} = 2, x_{2} = 2, x_{3} = 2$
Ways: $^{2} C_{2} \times^{4} C_{2} \times^{2} C_{2} = 1 \times 6 \times 1 = 6$

Calculating the Total Ways

Total Ways = Case 1 + Case 2 + Case 3 + Case 4
Total Ways = $4 + 8 + 8 + 6$
Total Ways = 26

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