- Option : B
- Explanation :
We must choose at least 3 women, so we calculate the case of 3 women, 4 women and 5 women and by addition rule add the result.

^{12}C_{3}x^{20}C_{2}+^{12}C_{4}x^{20}C_{1}+^{12}C_{5}x^{20}C_{0}= (12x11x10/3x2x1) x (20x19/2x1) + (12x11x10x9/4x3x2x1) x 20 + (12x11x10x9x8/5x4x3x2x1) x 1= 220 x 190 + 495 x 20 + 792

= 52492

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2. Which of the following statement(s) is/are false?

(a) A connected multigraph has an Euler Circuit if and only if each of its vertices has even degree.

(c) A complete graph (K_{n}) has a Hamilton Circuit whenever n ≥ 3

- Option : D
- Explanation :

EulerCircuits: An Euler circuit is a circuit that uses every edge of a graph exactly once.

HamiltonCircuit: Hamilton circuit, is a graph cycle (i.e., closed-loop) through a graph that visits each node exactly once

Bipartite Graph: A bipartite graph, also called a bigraph, is a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacent. A bipartite graph is a special case of a k-partite graph with k=2.

From the above definitions, we can see that (d) is false. So the answer is (D).

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- Option : C
- Explanation :

A simple graph G=(V,E) is called bipartite if its vertex set can be partitioned into two disjoint subsets V=V_{1}⋃V_{2}, such that every edge has the form e=(a,b) where aϵV_{1}and bϵV_{2}.

Bipartite graphs are equivalent to two-colorable graphs.

1. Assign Red color to the source vertex (putting into set V_{1}).

2. Color all the neighbours with Black color (putting into set V_{2}).

3. Color all neighbour’s neighbour with Red color (putting into set V_{1}).

4. This way, assign color to all vertices such that it satisfies all the constraints of m way coloring problem where m = 2.

5. While assigning colors, if we find a neighbour which is colored with same color as current vertex, then the graph cannot be colored with 2 colors (ie., graph is not Bipartite).

So answer is option (C).

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- Option : D
- Explanation :

Caterpillar tree: In graph theory, a caterpillar is a tree in which all the vertices are within distance 1 of a central path.

Theorem: All caterpillars are graceful.

So, (a), (b) and (c) are graceful.

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