Thirteen persons are sitting in a row. The number of ways in which four persons can be selected so that no two of them are consecutive is equal to

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The number of ways in which all the letters of the word “M A R R I A G E” can be permutated if no two vowels are together

The number of numbers lying between 100 and 1000 using only the digits 1,2,3,4,5,6,7 without repetition

The number of ways in which 4 alike chocolates can be distributed among 10 children so that each child gets at most one chocolate.

The number of triangles that can be formed by joining 12 points in a plane, of which exactly 5 are collinear

answer is B, C, D.

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Detailed Solution

$\begin{array}{l} x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 9, x_{1}, x_{5} \geq 0 \\ x_{2}, x_{3}, x_{4} \geq 1, number of solutions are 210 \end{array}$

$a) 5 \times 12 \times 12 = 720$

${b)}^{7} P_{3} = 210$

${c)}^{10} C_{4} = 210$

${d)}^{12} C_{3} -^{5} C_{3} = 210$

$\begin{aligned} x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 9, x_{1}, x_{5} \geq 0 \\ x_{2}, x_{3}, x_{4} \geq 1, number of solutions are 210 \end{aligned}$