If 5 dice are rolled simultaneously,then the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) To get a sum of $7$ with $5$ dice,each die must show at least $1$. Let the outcomes be $x_1, x_2, x_3, x_4, x_5$ where $x_i \ge 1$. The sum is $x_

Vedclass · Accepted Answer

$15$

Vedclass · Answer

(B) To get a sum of $7$ with $5$ dice,each die must show at least $1$. Let the outcomes be $x_1, x_2, x_3, x_4, x_5$ where $x_i \ge 1$. The sum is $x_1 + x_2 + x_3 + x_4 + x_5 = 7$. Since each $x_i \ge 1$,we can write $x_i = 1 + y_i$ where $y_i \ge 0$. Substituting this,we get $(1+y_1) + (1+y_2) + (1+y_3) + (1+y_4) + (1+y_5) = 7$,which simplifies to $y_1 + y_2 + y_3 + y_4 + y_5 = 2$. The number of non-negative integer solutions is given by the formula $\binom{n+r-1}{r-1}$ where $n=2$ and $r=5$. This is $\binom{2+5-1}{5-1} = \binom{6}{4} = \binom{6}{2} = \frac{6 	imes 5}{2} = 15$. Alternatively,the cases are: $(i)$ Four $1$s and one $3$: $\frac{5!}{4!1!} = 5$ ways. (ii) Three $1$s and two $2$s: $\frac{5!}{3!2!} = 10$ ways. Total ways = $5 + 10 = 15$.

If $5$ dice are rolled simultaneously,then the number of ways of getting a total of $7$ on their faces is:

Similar Questions

Out of $13$ cricket players,$4$ are bowlers. In how many ways can a team of $11$ players be selected such that the team contains at least $2$ bowlers?

There are $4$ oranges,$5$ apples,and $7$ mangoes in a fruit basket. The number of ways of selecting at least one fruit from among the fruits in the basket is

Six '$+$' and four '$-$' signs are to be placed in a straight line so that no two '$-$' signs come together. Then the total number of ways is:

$A$ team of $5$ students is to be selected from $12$ students. If two particular students are to be included in that team,then the number of ways that such team can be selected is

Vedclass Test Series

Exam Paper Generator

Online Exam Module