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NIMCET 2014 Mathematics PYQ — The number of ways in which days can be chosen in each of the mon… | Mathem Solvex

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NIMCET 2014 Mathematics PYQ — The number of ways in which days can be chosen in each of the mon… | Mathem Solvex | Mathem Solvex

Explanation

Solution

Concept:

There are $29$ days in the month of February in a leap year and $28$ days in a non-leap year. Other months have either $30$ or $31$ days.

The number of ways in which $r$ distinct objects can be selected from a group of $n$ distinct objects is given by the formula:

${}^{n}C_{r} = \frac{n!}{r!(n-r)!}$

Calculation:

In a non-leap year, the distribution of days in the $12$ months is as follows:

Month Number of Days

January, March, May, July, August, October, December $31$ days each ( $7$ months)

April, June, September, November $30$ days each ( $4$ months)

February $28$ days ( $1$ month)

Based on this:

There is 1 month with $28$ days. The number of ways to choose $5$ days is ${}^{28}C_5$ .

There are 4 months with $30$ days. The number of ways to choose $5$ days from each of these months is $({}^{30}C_5)^4$ .

There are 7 months with $31$ days. The number of ways to choose $5$ days from each of these months is $({}^{31}C_5)^7$ .

To find the total number of ways for all $12$ months, we multiply these values together:

$\text{Total ways} = ({}^{30}C_5)^4 \times ({}^{31}C_5)^7 \times ({}^{28}C_5)$

Correct Option: 1. $({}^{30}C_5)^4 \times ({}^{31}C_5)^7 \times ({}^{28}C_5)$

NIMCET 2014 — Mathematics PYQ

Choose the correct answer:

Explanation

Solution

Concept:

Calculation:

Explanation

Solution

Concept:

Calculation:

Month	Number of Days
January, March, May, July, August, October, December	$31$ days each ( $7$ months)
April, June, September, November	$30$ days each ( $4$ months)
February	$28$ days ( $1$ month)