Question

a lottery game requires that you pick 7 different numbers from 1 to 98. if you pick all 7 winning numbers, you win the jackpot. if you pick 6 of the 7 numbers correctly, you win $200,000. in how many ways can you pick exactly 6 of the 7 winning numbers without regard to order?
there are 644 ways to pick exactly 6 of the 7 winning numbers without regard to order.

Explanation:

Step1: Choose 6 winning numbers

We need to select 6 out of the 7 winning numbers. The number of ways to do this is given by the combination formula $C(n,k)=\frac{n!}{k!(n-k)!}$, where $n=7$ and $k=6$.
$C(7,6)=\frac{7!}{6!(7-6)!}=\frac{7!}{6!1!}=7$

Step2: Choose 1 non-winning number

We need to select 1 number from the non-winning numbers. There are $98-7=91$ non-winning numbers, so we choose 1 out of 91.
$C(91,1)=\frac{91!}{1!(91-1)!}=\frac{91!}{1!90!}=91$

Step3: Calculate total combinations

Multiply the number of ways from Step1 and Step2 to get the total number of ways to pick exactly 6 winning numbers.
$7\times91=637$

Answer:

637